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New Motifs
In DNA Nanotechnology

by
Nadrian C. Seeman, Hui Wang, Xiaoping Yang,
Furong Liu, Chengde Mao, Weiqiong Sun,
Lisa Wenzler, Zhiyong Shen, Ruojie Sha,
Hao Yan, Man Hoi Wong, Phiset Sa-Ardyen,
Bing Liu, Hangxia Qiu, Xiaojun Li, Jing Qi,
Shou Ming Du, Yuwen Zhang, John E. Mueller,
Tsu-Ju Fu, Yinli Wang, and Junghuei Chen

Department of Chemistry
New York University
New York, NY 10003, USA
Ned.Seeman@NYU.edu

This is a draft paper for a talk at the
Fifth Foresight Conference on Molecular Nanotechnology.
The final version has been submitted
for publication in the special Conference issue of
Nanotechnology.

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Abstract

In recent years, we have invested a great deal of effort to construct molecular building blocks from unusual DNA motifs. DNA is an extremely favorable construction medium: The sticky-ended association of DNA molecules occurs with high specificity, and it results in the formation of B-DNA, whose structure is well known. The use of stable branched DNA molecules permits one to make stick-figures. We have used this strategy to construct a covalently closed DNA molecule whose helix axes have the connectivity of a cube, and a second molecule, whose helix axes have the connectivity of a truncated octahedron.

In addition to branching topology, DNA also affords control of linking topology, because double helical half-turns of B-DNA or Z-DNA can be equated, respectively, with negative or positive crossings in topological objects. Consequently, we have been able to use DNA to make trefoil knots of both signs and figure-8 knots. By making RNA knots, we have discovered the existence of an RNA topoisomerase. DNA-based topological control has also led to the construction of Borromean rings, which could be used in DNA-based computing applications.

The key feature previously lacking in DNA construction has been a rigid molecule. We have discovered that DNA double crossover molecules can provide this capability. We have incorporated these components in DNA assemblies that make use of this rigidity to achieve control on the geometrical level, as well as on the topological level. Some of these involve double crossover molecules, and others involve double crossovers associated with geometrical figures, such as triangles and deltahedra.

Introduction

DNA is well-known as the polymeric molecule that contains the genetic information for life. Its key chemical feature is its ability to associate with and recognize other DNA molecules by means of specific base pairing relationships: Thus, an adenine (A) on one strand will pair preferentially with a thymine (T) on the other strand; likewise, guanine (G) will pair with cytosine (C). This complementary relationship has been known for about 45 years as the chemical basis for heredity [1]. Since the early 1970's, genetic engineers have been using a variation on this theme to associate specific DNA double helices with each other [2]. As shown in Figure 1, a double helix with a single-stranded overhang (often called a 'sticky end') will hydrogen bond with a complementary overhang to bring two DNA molecules into proximity; Figure 1 also shows that if desired the two pieces of DNA can be joined covalently to form a single double helix.

Fig. 1. Sticky Ended Association

Figure 1. Sticky-Ended Cohesion and Ligation. Two linear double helical molecules of DNA are shown at the top of the drawing. The antiparallel backbones are indicated by the black lines terminating in half-arrows. The half-arrows indicate the 5'-->3' directions of the backbones. The right end of the left molecule and the left end of the right molecule have single-stranded extensions ('sticky ends') that are complementary to each other. The middle portion shows that, under the proper conditions, these bind to each other specifically by hydrogen bonding. The bottom of the drawing shows that they can be ligated to covalency by the proper enzymes and cofactors.

Assemblies involving traditional linear double helical pieces of DNA correspond to the concatenation of line segments. However, it is possible to design and assemble sequences of synthetic DNA molecules that form stable branches (called 'junctions') flanked by 3-6 arms [3]. The same logic applies to the association of branched molecules that applies to linear molecules. However, by using branched molecules, it is possible to form stick figures whose connectivity is no longer trivial. An example of this type of construction is illustrated in Figure 2. In this regard, we have previously reported the construction of a cube [4], shown in Figure 3, and a truncated octahedron [5], shown in Figure 4. The edges of each of these stick polyhedra are composed of double helical DNA.

Fig. 2. Branched Junction Assembly

Figure 2. Formation of a Two-Dimensional Lattice from an Immobile Junction with Sticky Ends. A is a sticky end and A' is its complement. The same relationship exists between B and B'. Four of the monomeric junctions on the left are complexed in parallel orientation to yield the structure on the right. A and B are different from each other, as indicated by the pairing in the complex. Ligation by DNA ligase can close the gaps left in the complex. The complex has maintained open valences, so that it could be extended by the addition of more monomers.

Fig. 3. A DNA Cube

Figure 3. A DNA Molecule with the Connectivity of a Cube. This representation of a DNA cube shows that it contains six different cyclic strands. Each nucleotide is represented by a single colored dot for the backbone and a single white dot representing the base. Note that the helix axes of the molecule have the connectivity of a cube. However, the strands are linked to each other twice on every edge. Therefore, this molecule is a hexacatenane. To get a feeling for the molecule, follow the front strand around its cycle: It is linked twice to to each of the four strands that flank it, and only indirectly to the strand at the rear. Note that each edge of the cube is a piece of double helical DNA, containing two turns of the double helix.

Fig. 4. A DNA Truncated Octahedron

Figure 4. A DNA Molecule with the Connectivity of a Truncated Octahedron. A truncated octahedron contains six squares and eight hexagons. This is a view down the fourfold axis of one of the squares. Each edge of the truncated octahedron contains two double helical turns of DNA. The molecule contains 14 cyclic strands of DNA. Each face of the octahedron corresponds to a different cyclic strand. In this drawing, each nucleotide is shown with a colored dot corresponding to the backbone, and a white dot corresponding to the base. This picture shows the strand corresponding to the square at the center of the figure and parts of the four strands at the cardinal points of the figure. In addition to the 36 edges of the truncated octahedron, each vertex contains a hairpin of DNA extending from it. These hairpins are all parts of the strands that correspond to the squares. The molecular weight of this molecule as about 790,000 Daltons.

In this article, we will first summarize the properties of DNA as a construction material. We will review briefly the techniques for the construction and demonstration of DNA polyhedra. Next, we will describe the relationships that act as the basis for the construction of DNA knots and catenanes. Finally, we will discuss the search for rigid DNA motifs, and the means to incorporate them into DNA nanotechnology.

DNA as a Construction Material

There are several advantages to using DNA for nanotechnological constructions. First, the ability to get sticky ends to associate makes DNA the molecule whose intermolecular interactions are the most readily programmed and reliably predicted: Sophisticated docking experiments needed for other systems reduce in DNA to the simple rules that A pairs with T and G pairs with C. In addition to the specificity of interaction, the local structure of the complex at the interface is also known: Sticky ends associate to form B-DNA [6]. A second advantage of DNA is the availability of arbitrary sequences, due to convenient solid support synthesis [7]. The needs of the biotechnology industry have also led to straightforward chemistry to produce modifications, such as biotin groups, fluorescent labels, and linking functions. The recent advent of parallel synthesis [8] is likely to increase the availability of DNA molecules for nanotechnological purposes. DNA-based computing [9] is another area driving the demand for DNA synthetic capabilities. Third, DNA can be manipulated and modified by a large battery of enzymes, including DNA ligase, restriction endonucleases, kinases and exonucleases. In addition, double helical DNA is a stiff polymer [10] in 1-3 turn lengths, it is a stable molecule, and it has an external code that can be read by proteins and nucleic acids [11].

There are two properties of branched DNA that one cannot ignore: First, the angles between the arms of branched junctions are variable. In contrast, to the trigonal or tetrahedral carbon atom, ligation-closure experiments [12], [13] have demonstrated branched junctions are not well-defined geometrically. Thus, the cube and the truncated octahedron discussed above are molecules whose graphs correspond to the graphs of those ideal objects (e.g., [14]), but only their branching connectivity has been (or probably can be) demonstrated. Simple branched junctions apparently do not lead to geometrical control. This places a greater burden on specificity: The construction illustrated in Figure 2 would not lead exclusively to the quadrilateral depicted there unless the inter-arm angles were fixed to be right angles. Nevertheless, it is possible to generate a quadrilateral by using four different sticky end pairs to make each of the four edges [15].

Second, it is imperative to recognize that DNA is a helical molecule. For many purposes, the double helical half-turn is the quantum of single-stranded DNA topology. Figure 5 illustrates two variants of Figure 1, one with an even number of half-turns between vertices, and the other with an odd number. With an even number of half-turns, the underlying substructure is a series of catenated single-stranded cycles, much like chain-mail, but an odd number leads to an interweaving of long strands. If the edges flanking a face of a polyhedron contain an exact number of helical turns, then that face contains a cyclic strand as one of its components; this strand will be linked (in the topological sense) to the strands of the adjacent faces, once for every turn in their shared edges. We used this design motif with both the cube and the truncated octahedron, so they are really a hexacatenane and a 14-catenane. In general, the level control over linking topology available from DNA is almost equal to the level of control over branching topology. Consequently, a number of topological species have been constructed relatively easily from DNA, even though they represented extremely difficult syntheses using the standard tools of organic and inorganic chemistry.

Fig. 5. Topological Assembly

Figure 5. Topological Consequences of Ligating DNA Molecules Containing Even and Odd Numbers of DNA Half-Turns in Each Edge. These diagrams represent the same ligation shown in Figure 2. However, they indicate the plectonemic winding of the DNA, and its consequences. The DNA is drawn as a series of right-angled turns. In the left panel, each edge of each square contains two turns of double helix. Therefore, each square contains a cyclic molecule linked to four others. In the right panel, each edge of each square contains 1.5 turns of DNA. Therefore, the strands do not form cycles, but extend infinitely in a warp and weft meshwork.

The Construction and Analysis of DNA Polyhedra

The combination of branched DNA and sticky-ended ligation results in the ability to form stick figures whose edges consist of double helical DNA, and whose vertices are the branch points of the junctions. The flexibility of the angles that flank the branch points of junctions results in the need to specify connectivity explicitly. This may be done either by a set of unique sticky end pairs, one for each edge [4], [15], or by utilizing a protection-deprotection strategy [16] so that only a given pair is available for ligation at a particular moment. The first strategy was used in the construction of the DNA cube, which was done in solution [4].

We found that we had too little control over the synthesis when it was done in solution, so we developed a solid-support-based methodology [16]. This approach allows convenient removal of reagents and catalysts from the growing product. Each ligation cycle creates a robust intermediate object that is covalently closed and topologically bonded together. The method permits one to build a single edge of an object at a time, and to perform intermolecular ligations under conditions different from intramolecular ligations. Control derives from the restriction of hairpin loops forming each side of the new edge, thus incorporating the technique of successive deprotection. Intermolecular reactions are done best with asymmetric sticky ends, to generate specificity. Sequences are chosen in such a way that restriction sites are destroyed when the edge forms. One of the major advantages of using the solid support is that the growing objects are isolated from each other. This permits the use of symmetric sticky ends, without intermolecular ligation occurring. More generally, the solid support methodology permits one to plan a construction as though there were only a single object to consider. Many of the differences between a single molecule and a solution containing 1012 molecules disappear if the molecules are isolated on a solid support. We utilized the solid-support methodology to construct the DNA truncated octahedron.

The polyhedra we made were objects that were topologically specified, rather than geometrically specified; consequently, our proofs of synthesis were also proofs of topology. In each case, we incorporated restriction sites in appropriate edges of the objects, and then broke them down to target catenanes, whose electrophoretic properties could be characterized against standards [17]. For example, the first step of synthesizing the cube resulted in the linear triple catenane corresponding to the ultimate left-front-right sides of the target. When the target was achieved, one of the most robust proofs of synthesis came from the restriction of the two edges in the starting linear triple catenane, to yield the linear triple catenane corresponding to the top-back-bottom of the cube, as shown in Figure 6. A similar approach was taken with the proof of the truncated octahedron synthesis: The presence of the six square strands was demonstrated first. Then the octacatenane corresponding to the eight hexagonal faces was shown by restricting it down to the tetracatenane flanking each square, for which we were able to make a marker.

Fig. 6. The Cube as a Sum of Linear Catenanes

Figure 6. The Linear Triple Catenanes that Link to Form the Cube. The target cube is shown at the left of the figure. The starting material for its synthesis was the linear triple catenane shown at the center of the drawing. This catenane corresponds to the left, front and right faces of the cube. When the cube is restricted on its two front edges, the starting linear triple catenane is destroyed. However, when the cube is successfully synthesized, a linear triple catenane results. This catenane corresponds to the top, back and bottom faces of the cube.

The solid-support based methodology appears to be quite powerful. We feel that we could probably construct most Platonic, Archimedean, Catalan, or irregular polyhedra by using it. The cube is a 3-connected object, as is the truncated octahedron. The cube was constructed from 3-arm branched junctions, but the truncated octahedron was constructed from 4-arm branched junctions, because we had originally planned to link the truncated octahedra together. The connectivity [18], [19] of an object or a network determines the minimum number of arms that can flank the junctions that act as its vertices. Thus, one must have at least 5-arm branched junctions to construct an icosahedron, and one must have 12-arm branched junctions to build a cubic-close-packed (face-centered cubic) lattice. We have built junctions with up to 6 arms [3], but there seem to be no impediments to making junctions containing arbitrary numbers of arms. The one caveat to observe is that the lengths of the arms necessary for stabilization tend to increase with the number of arms.

Topological Construction

In the last section, we have emphasized that the construction of DNA polyhedra ultimately becomes an exercise in synthetic topology: The resulting structures are characterized best by their branching and linking rather than by their geometry. In addition to the construction of polyhedral catenanes, DNA nanotechnology is also an extremely powerful methodology for the construction of knots, unusual links, and other species defined by their linking. Indeed, it is arguably the most powerful system for creating these targets.

The key requisite for constructing topological targets is the ability to produce at will a chemical version of a node or a crossing (sometimes called a unit tangle) in the target. The strength of DNA in this regard derives from the fact that a half-turn of DNA corresponds exactly to this necessary component [20]. It is easy to understand this relationship by looking at Figure 7. Here, a trefoil knot has been drawn, with an arbitrary polarity. Squares have been placed about each of the crossings, so that the portions of the knot contained within each square act as its diagonals. These diagonals divide the square into four regions, two between parallel strands, and two between antiparallel strands. Whereas the strands of double helical DNA are antiparallel, one should design the sequence of the DNA strand so that pairing occurs over a half-turn segment (ca. 6 nucleotide pairs) in the regions between antiparallel strands. Thus, it is possible to make the transition from topology to nucleic acid chemistry by specifying complementary sequences to form desired nodes. Linker regions between the nodes usually consist of oligo-dT.

Fig. 7. Nodes as Half-Turns of Double Helical DNA

Figure 7. The Relationship Between Nodes and Antiparallel B-DNA Illustrated on a Trefoil Knot. A trefoil knot is drawn with negative nodes. Nodes are also known as crossings or unit tangles. The path is indicated by the arrows and the very thick curved lines connecting them. The nodes formed by the individual arrows are drawn at right angles to each other. Each pair of arrows forming a node defines a quadrilateral (a square in this figure), which is drawn in dotted lines. Each square is divided by the arrows into four domains, two between parallel arrows and two between antiparallel arrows. The domains between antiparallel arrows contain lines that correspond to base pairing between antiparallel DNA (or RNA) strands. Dotted double-arrowheaded helix axes are shown perpendicular to these lines. The twofold axis that relates the two strands is perpendicular to the helix axis; its ends are indicated by lens-shaped figures. The twofold axis intersects the helix axis and lies halfway between the upper and lower strands. The amount of DNA shown corresponds to about half a helical turn. It can be seen that three helical segments of this length could assemble to form a trefoil knot. The DNA shown could be in the form of a 3-arm DNA branched junction. A trefoil of the opposite sense would need to be made from Z-DNA, in order to generate positive nodes.

There are two kinds of nodes found in topological species, positive nodes and negative nodes. As illustrated at the top of Figure 8, these nodes are mirror images of each other. B-DNA is a right-handed helical molecule. Its crossings generate nodes that are designated to have negative signs, as illustrated at the bottom-left side of the drawing. Fortunately, there is another form of DNA, Z-DNA, shown at the bottom-right, whose helix is left-handed [21]. Z-DNA is not the geometrical mirror image of B-DNA, because it still contains D-deoxyribose sugar residues, and, in addition, its structure is qualitatively different. However, from a topological standpoint, it is the mirror image of B-DNA, and it can be used to supply positive nodes when they are needed.

Fig. 8. Node Chirality

Figure 8. Nodes and DNA Handedness. The upper part of this drawing shows positive and negative nodes, with their signs indicated. It is useful to think of the arrows as indicating the 5'-->3' directions of the DNA backbone. Below the negative node is a representation of about one and a half turns of a right-handed B-DNA molecule. Note that the nodes are all negative. Below the positive node is a left-handed DNA molecule, termed Z-DNA. The Z-DNA molecule has a zig-zag backbone, which we have tried to indicate here. However, the zig-zag nature of the backbone does not affect the fact that all the nodes are positive.

The Z-forming propensity of a segment of DNA is a function of two variables, the sequence, and the conditions. Not all sequences undergo the B-->Z transition under the mild conditions compatible with enzymatic ligation. The sequence of conventional nucleotides that undergoes the transition most readily contains the repeating dinucleotide sequence dCdG. Furthermore, the ease with which a segment undergoes the B-->Z transition can be made a function of base modification; DNA in which a methyl group has been added to the 5-position of cytosine undergoes the transition under milder conditions [21]. However, in the absence of Z-promoting conditions, the sequence will remain in the B-form.

We have utilized this basic framework to construct a number of knotted species from DNA molecules. Figure 9 illustrates a molecule with two pairing domains, each containing one turn of DNA double helix. Each of the two domains is capable of undergoing the B-->Z transition, but one of the domains undergoes the transition more readily than the other one. At very low ionic strength, neither domain forms double helical DNA, and a molecule with circular topology results. At higher ionic strength, both domains form B-DNA, and a trefoil knot results, with all of its nodes negative. Under mild Z-promoting conditions, the more sensitive domain converts to Z-DNA, and a figure-8 knot is the product. When the solution presents more vigorous Z-promoting conditions, the other domain also converts to Z-DNA, and ligation yields the trefoil knot with positive nodes [22].

Fig. 9. A DNA Strand in Four Topological States

Figure 9. A DNA Strand is Ligated into Four Topological States by Variation of Ligation Conditions. The left side of this synthetic scheme indicates the molecule from which the target products are produced. The four pairing regions, X and its complement X', Y and its complement Y' are indicated by the bulges from the square. The 3' end of the molecule is denoted by the arrowhead. The four independent solution conditions used to generate the target products are shown to the right of the basic structure. The pairing and helical handedness expected in each case is shown to the right of these conditions, and the molecular topology of the products is shown on the far right of the figure. The species are, from the top, the circle, the trefoil knot with negative nodes, the figure-8 knot, and the trefoil knot with positive nodes.

The favored topology of each of the species in Figure 9 is a function of solution conditions. If one of these molecules is placed in solution conditions that favor one of the other knots, it cannot convert to the new favored structure without breaking and rejoining its backbone. However, type I DNA topoisomerases can catalyze this interconversion [23]. Figure 10 illustrates the stepwise interconversion of the different species, under solution conditions that promote the B-->Z or Z-->B transitions.

Fig. 10. Interconversion of DNA Knots

Figure 10. DNA Knots Interconverted by Type I DNA topoisomerases. On the top of this figure are the three knots that are interconverted, the trefoil knot with positive nodes, The figure-8 knot, and the trefoil knot with negative nodes. The nucleotide pairs that give rise to the nodes are indicated between strands. The same knots are shown in the bottom portion of the figure, interspersed by circles drawn with the node structures of dumbbells. The lines indicating the base pairs have been removed for clarity. The '+' and '-' signs near the nodes indicate their topological signs. The equilibria indicated between structures are catalyzed by the E. coli DNA Topoisomerases I and III. The trefoil knot on the left has all positive signs, and the signs of a single node at a time are switched from positive to negative in each of the structures as one proceeds towards the right of the figure. Changing the sign of a single node in the positive trefoil knot produces a circle (dumbbell), and changing a second node in the same domain produces a figure-8 knot. Changing the sign of another positive node in the figure-8 knot produces the circle (dumbbell) on the right, and changing the sign of the last node generates the negative trefoil knot. It is important to realize that the two circles shown may interconvert without the catalytic activity of a topoisomerase.

This ability of topoisomerases to interconvert synthetic DNA knots suggested to us that it would be possible to use an RNA knot to assay the presence of an RNA topoisomerase, a species unknown previously. By preparing both an RNA knot and an RNA circle, we found that it was possible to catalyze the interconversion of these cyclic molecules by the presence of E. coli DNA topoisomerase III [24]. This experiment is illustrated in Figure 11.

Fig. 11. Discovery of an RNA Topoisomerase

Figure 11. The Discovery of an RNA Topoisomerase An RNA single strand is shown at the top of this diagram. Its Watson-Crick pairing regions, X, Y, X' and Y' are illustrated at bumps on the square, and the spacers, denoted by S are shown as the corners of the square. The arrowhead denotes the 3' end of the strand. The pathway to the left illustrates formation of the RNA circle: A 40 nucleotide DNA linker (incompatible with knot formation) is annealed to the molecule, and it is ligated together to form an RNA circle, which survives treatment with DNase. In the other pathway, a 16 nucleotide DNA linker is used in the same protocol to produce the RNA trefoil knot, whose three negative nodes are indicated. The interconversion of the two species by E. coli DNA Topoisomerase III (Topo III) is shown at the bottom of the figure. The 40-mer RNA strand promotes somewhat the formation of the circle from the knot. E. coli DNA Topoisomerase I does not catalyze this reaction.

In order to illustrate the power of DNA as a medium for the assembly of topological targets, we have recently used this system to construct Borromean rings from DNA [25]. Borromean rings are a rich family of topological structures [26] whose simplest member (section [a] of Figure 12) appears on the coat of arms of the Borromeo family, prominent in the Italian Renaissance. Their key property is that removal of any individual circle unlinks the remaining rings. The innermost three nodes are negative, and the outermost three are positive. Although it is possible to fashion topological targets from DNA molecules held together by a single half-turn of DNA [27], it is often more convenient to use 1.5 turns of DNA, if this does not change any key features of the target. Therefore, we converted the traditional Borromean ring structure to one that replaced each crossing with three crossings (part [b] of Figure 12). It is evident that the innermost three segments correspond to a 3-arm DNA branched junction made from B-DNA.

Fig. 12. Borromean Rings

Figure 12. The Design and Construction of Borromean Rings from DNA

[a] Traditional Borromean Rings. Borromean rings are special links, because linkage between any pair of rings disappears in the absence of the third. The signs of the three nodes near the center of the drawing are negative, and the signs of the outer three nodes are positive.

[b] Borromean Rings with Each Node Replaced by Three Nodes. Each node of [a] has been replaced by three nodes, derived from 1.5 turns of DNA double helix. The inner double helices are right handed, corresponding to B-DNA, and the outer double helices are left handed, corresponding to Z-DNA. Think of this drawing like a polar projection of the Earth, where the center is at the North Pole, and every point on the circumference corresponds to the South Pole.

[c] Stereoscopic Representation of [b]. View this picture with stereo glasses, or you can learn to see stereo by diverging your eyes. The 'projection' of [b] is represented in 3-D, now. The three outer double helices have been folded under the inner double helices, so that a B-DNA 3-arm branched junction flanks the 'North Pole' of the object and a Z-DNA 3-arm branched junction flanks the 'South Pole' of the object.

[d] Stereoscopic Views of the DNA Molecules Synthesized. Two hairpins have been added to the 'equatorial' sections of each strand. Each hairpin contains a site for a restriction endonuclease, so that the Borromean property can be demonstrated in the test tube.

With a topological picture, it is always permissible to deform it. One can imagine that this picture corresponds to a polar map of the earth, where the center is at the North Pole, and every point on the circumference represents the South Pole. Thus, the three points at the outermost radii of the three helices could all abut each other at the South Pole. Section [c] of Figure 12 is a stereoscopic view that illustrates what this molecule would look like if it were wrapped around a sphere. From this view, it is clear that the three outermost helices represent a 3-arm branched junction made from Z-DNA. From both synthetic and analytical standpoints, it is convenient to have a series of hairpins at 'the equator', as illustrated in section [d] of Figure 12. We have been able to use them as sites both to ligate the two junctions together, and to restrict them. By designing them to be slightly different lengths, it is easy to separate the restriction products on a gel.

Our ability to construct Borromean rings demonstrates that the 3-D geometrical approach we used has facilitated the exploitation of the relationship between nodes and DNA half-turns. This scheme consists of {1} identifying components to serve as positive and negative nodes (or their odd multiples), {2} linking components in a minimal number of spatially condensed stable units (3-arm branched junctions here), followed by {3} recognition-directed ligation; this approach should provide topological control in other chemical systems. Conversely, it may be possible to use this or other successful systems to act as scaffolding that guides the formation of target topological products from other polymers.

Besides being a holy grail for synthetic chemistry, Borromean rings might be able to serve a role in DNA-based computing. It is possible to design Borromean rings that contain an arbitrary number of circles, so they are not limited to just three strands. A complete Borromean complex can be separated readily from its dissociated components. It is not hard to imagine that the integrity of a Borromean link can represent the truth of each of a group of logical statements. If any one of them is false, then one of the rings would not be closed. From a chemical point of view, these two cases would be separated easily by denaturing gel electrophoresis. For example, one could use the integrity of a Borromean link as a check that the right molecules had associated, in a set of interactions orthogonal to the main calculation. In this capacity, the presence of the Borromean link would function as parity-checking did on early computers: If the calculation has been done right, the link is established, and otherwise it is broken, and those molecules lacking an intact link could be discarded.

The Quest for Rigidity

We have emphasized above the power of the solid-support based synthetic approach to DNA nanotechnology. It allows us to construct discrete objects containing a finite number of edges. However, one of the key goals of DNA nanotechnology is the ability to construct precisely configured materials on a much larger scale. A particularly important goal in this regard is the assembly of periodic matter, namely crystals [28]; this ability offers both a window on the crystallization problem for macromolecules, [28] and on the assembly of molecular electronic components [29]. Periodic matter entails a whole new series of problems. The strength of DNA nanotechnology is that the specificity of intermolecular interactions can be used to make defined objects. In particular, the ability to program different sticky ends to form the edges of a polyhedron or other target gives us a tremendous amount of control over the product. Another way to say this is that we have used an asymmetric set of sticky ends, because none of them are the same. The key to control over the products of a reaction is the minimization of symmetry. Symmetry is antithetical to control.

However, when we wish to make crystalline materials, we are forced to consider the case where symmetry dominates. The distinguishing characteristic of crystals is their translational symmetry: The contacts on the left side of a crystalline unit cell must complement those on the right side in an infinite array; the top and bottom, and the front and the rear bear the same relationship. It is very hard to achieve an infinite arrangement with flexible components. The reason is that flexible components do not maintain the same spatial relationships between each member of a set. Consequently, instead of periodic matter, one often obtains a random network. In addition, a flexible system can cyclize on itself, thereby poisoning growth. Hence, it is key for the success of building periodic matter to discover rigid DNA components.

Recognition of this situation has led us to two different complementary approaches in the quest for rigidity. The first of these is to abandon potentially flexible polygonal and polyhedral motifs. A theory of bracing such systems exists (e.g., [14]), but it is simplest to restrict ourselves to triangles and deltahedra (polygons whose faces are all triangles). A convex polyhedron can be shown to be rigid if and only if its faces are exclusively triangular [14]. The second approach has been to seek rigid DNA motifs. We have investigated the flexibility of bulged DNA branched junctions. Initially, they seemed promising because they were stiffer than conventional junctions [30]. Ultimately, however, they did not bear up to rigorous testing [31]. Fortunately, we have discovered another motif, the antiparallel DNA double crossover molecule [32], that appears to be far stiffer than bulged junctions [33].

DNA double crossover molecules (abbreviated DX) are analogs of intermediates in the process of genetic recombination [34]. They correspond to pairs of 4-arm branched junctions that have been ligated at two adjacent arms. We have used them extensively to explore the properties of conventional branched junctions, including their susceptibility to enzymes [35], their crossover topology [36], and their crossover isomerization [37], [38]; we have also used them to make symmetric immobile branched junctions [39]. Figure 13 shows that there are five possible isomers of DX molecules. Three of them contain parallel helical domains (DPE, DPOW and DPON), and two contain antiparallel helical domains (DAE and DAO). Those with the parallel domains are relevant to biological processes, but those with antiparallel domains are far more stable in systems with a small separation between the crossovers. The difference between DAE and DAO is the number of double helical half-turns between crossovers, an even number (DAE) or an odd number (DAO). The two odd parallel DX molecules differ by whether the extra half-turn is a wide groove (DPOW) or narrow groove (DPON) segment; this issue does not arise in antiparallel DX molecules.

Fig. 13. DNA Double Crossover Molecules

Figure 13. The Isomers of DNA Double Crossover Molecules. The structures shown are named by the acronym describing their basic characteristics. All names begin with 'D' for double crossover. The second character refers to the relative orientations of their two double helical domains, 'A' for antiparallel and 'P' for parallel. The third character refers to the number (modulus 2) of helical half-turns between crossovers, 'E' for an even number and 'O' for an odd number. A fourth character is needed to describe parallel double crossover molecules with an odd number of helical half-turns between crossovers. The extra half-turn can correspond to a major (wide) groove separation, designated by 'W', or an extra minor (narrow) groove separation, designated by 'N'. The strands are drawn as zig-zag helical structures, where two consecutive, perpendicular lines correspond to a full helical turn for a strand. The arrowheads at the ends of the strands designate their 3' ends. The structures contain implicit symmetry, which is indicated by the conventional markings, a lens-shaped figure (DAE) indicating a potential dyad perpendicular to the plane of the page, and arrows indicating a twofold axis lying in the plane of the page. Note that the dyad in DAE is only approximate, because the central strand contains a nick, which destroys the symmetry. The strands have been drawn with pens of two different colors (three for DAE), as an aid to visualizing the symmetry. In the case of the parallel strands, the red strands are related to the other red strands by the twofold axes vertical on the page; similarly, the blue strands are symmetrically related to the blue strands. The twofold axis perpendicular to the page (DAE) relates the two red helical strands to each other, and the two blue outer crossover strands to each other. The 5' end of the central green double crossover strand is related to the 3' end by the same dyad element. A different convention is used with DAO. Here, the blue strands are related to the red strands by the dyad axis lying horizontal on the page. An attempt has been made to portray the differences between the major and minor grooves. Note the differences between the central portions of DPOW and DPON. Also note that the symmetry brings symmetrically related portions of backbones into apposition along the center lines in parallel molecules, in these projections. The same contacts are seen to be skewed in projection for the antiparallel molecules.

Our usual means for assaying rigidity is a ligation-closure experiment. Figure 14 illustrates such an experiment for a 3-arm branched junction. The products are assayed to see whether oligomerization has led to cyclization, and, if so, whether there is a single product or a collection of them. A collection of cyclic products suggests that the angles between the arms of the molecule being tested are not well-fixed. A key feature of this experiment is that the oligomerized species must contain an accessible 'reporter strand', whose fate is the same as that of the complex. Figure 15 illustrates the topological consequences of ligating DAE and DAO molecules; only the DAE molecule generates a reporter strand. The DAE molecule contains 5 strands (in contrast to 4 strands in a DAO molecule), and the central strand is often difficult to seal shut. However, another option is to extend it as a bulged 3-arm junction. Figure 15 shows that ligation of this molecule (DAE+J) also generates a reporter strand. Ligation of both DAE and DAE+J result in negligible amounts of cyclization: A small amount is detected for DAE+J, but none is seen for DAE.

Fig. 14. Reporter Strands in Ligation-Closure Experiments

Figure 14. Reporter Strands in Ligation-Closure Experiments. The 3-arm junction employed is indicated at the upper left of the diagram. The 3' ends of the strands are indicated by half-arrowheads. The 5' end of the top strand contains a radioactive phosphate, indicated by the starburst pattern, and the 5' end of the strand on the right contains a non-radioactive phosphate, indicated by the filled circle. The third strand corresponds to the blunt end, and is not phosphorylated. Beneath this molecule are shown the earliest products of ligation, the linear dimer, the linear trimer and the linear tetramer. The earliest cyclic products are shown on the right, the cyclic trimer and the cyclic tetramer. The blunt ends form the exocyclic arms of these cyclic molecules. Note that in each case the labeled strand has the same characteristics as the entire complex: It is an oligomer of the same multiplicity as the complex, and its state of cyclization is that of the complex. Hence, it can function as a reporter strand When the reaction is complete, the reaction mixture is loaded onto a denaturing gel, and its autoradiogram is obtained. Both cyclic and linear products are found, as indicated on the left of the gel. If an aliquot of the reaction mixture is treated with exo III and/or exo I, the linear molecules are digested, and only the cyclic molecules remain. Not shown in this cartoon are the linear and cyclic markers also run on the gel, so that the strands can be sized absolutely.

Fig. 15. Antiparallel Double Crossover Ligation

Figure 15. The Products of Antiparallel Double Crossover Ligation. Shown at the top of the diagram are three types of antiparallel double crossover molecules, DAE, with an even number of double helical half-turns between the crossover, DAO, with an odd number of half-turns between the crossovers, and DAE+J, similar to DAE, but with a bulged junction emanating from the nick in the central strand. The DAE and DAE+J molecules contain 5 strands, two of which are continuous, or helical strands, and three of which are crossover strands including the cyclic strands in the middle. The 3' ends of each strand are indicated by an arrowhead. The DAO molecule contains only 4 strands. The twofold symmetry element is indicated perpendicular to the page for the DAE molecule, and it is horizontal within the page for the DAO molecule. The drawing below these diagrams represents DAE, DAO and DAE+J molecules in which one helical domain has been sealed by hairpin loops, and then the molecules have been ligated together. The ligated DAE and DAE+J molecules contain a reporter strand. By contrast, the ligated DAO molecule is a series of catenated molecules.

This motif is significantly different from the single branched junction motif, and we have to figure out how to use it, particularly in combination with triangles and deltahedra. Figure 16 shows a series of double crossover molecules oriented to form a trigonal set of vectors by means of their attachment to a triangle. The triangles are connected, so as to tile a plane. Thus, it appears possible to use DAE molecules to form a two dimensional DNA lattice. In our hands, DAO molecules are usually better behaved than DAE molecules, so it is likely that they can be used even more effectively than DAE molecules for this purpose, so long as a reporter strand is not needed to ascertain the results of the construction.

Fig. 16. DNA Double Crossover Triangle Lattice

Figure 16. A Two-Dimensional Lattice Formed from Triangles Flanked by Double Crossover Molecules This diagram shows a series of equilateral triangles whose sides consist of double crossover molecules. These triangles have been assembled into an hexagonally-symmetric two-dimensional lattice. The basic assumption here is that triangles will retain their angular distributions here, so that they represent eccentric trigonal valence clusters of DNA.

We have tested whether it is possible for a double crossover molecule to be attached to a triangular motif and still maintain its structural integrity. Figure 17 illustrates an experiment in which two DNA double crossover molecules have been used to form the sides of a DNA triangle. The domains that form the sides of the triangle correspond to the domains in Figure 15 that were capped with hairpins. The other domains have been ligated to oligomerize the structure, either the domain at the bottom, or the domain on the left side, in two separate experiments. In both cases, linear reporter strands are recovered, and no cyclic reporter strands are detected. Thus, it is possible to incorporate DX molecules into the sides of a triangle, and to maintain their structural integrity.

Fig. 17. Ligation of a Triangle With Two DX Edges

Figure 17. A Ligation Experiment Using a Triangle With Two DX Edges. The triangle shown at the top contains two DAE double crossover molecules in its edges. In the experiment shown, one of them has biotin groups on each of its hairpins. When the triangle is restricted, to unmask sticky ends, restriction may not be complete. Molecules that have been properly restricted will contain no biotins, but those with incomplete restriction will have a biotin attached. These incompletely restricted molecules can be removed by treatment with streptavidin beads. The purified triangles with sticky ends can be ligated together. There is no evidence of cyclization in the reporter strands produced by this experiment. The representation of the DNA as a ladder makes it appear that there are no reporter strands, but this is not the case, when the DNA is drawn as a double helix.

Figure 18 illustrates a means of utilizing DAE+J molecules to form a lattice. This figure shows the same lattice employed in Figure 16. However, the extra junction is used to form the triangles, and the other domain of the double crossover molecule is used to buttress the edge and to keep its helix axis linear.

Fig. 18. DNA+J Triangle Lattice

Figure 18. A Triangle Lattice Formed from the DAE+J Motif. The DAE+J molecules used here serve to buttress branched junctions, to keep them from bending. The triangles are formed using the extra junction, so that it is part of the lattice, in contrast to the triangular lattice formed from simple DAE molecules, shown in Figure 16. Exactly the same arrangement of triangles has been employed here.

Figure 19 shows the extension to three dimensions of the scheme illustrated in Figure 16. A single octahedron is drawn, containing three double crossover molecules. The free helical domains of these DX edges span a three-dimensional space, and they will not intersect each other, no matter how far they are extended. An enantiomorphous set of three arms could also be chosen. If each of the three arms were connected to its corresponding arm in another octahedron, the resulting structure would nucleate an array resembling the arrangement of octahedral subunits in cubic close packed structures (face-centered cubic structures). However, the structure would be of lower symmetry, because of the connections through the outer helical domains. Figure 20 shows a schematic representation of the components of this rhombohedral system. Figure 21 shows a view down the 3-fold axis of the array.

Fig. 19. A DNA Octahedron Flanked by Double Crossover Molecules

Figure 19. An Octahedron Containing Three Edges Made from Double Crossovers. This drawing of an octahedron down one of its three-fold axes shows only four of its eight equilateral triangular faces. The three edges shown constructed from DAE molecules are not coplanar, but span a three-dimensional space. An enantiomorphous set also exists. Connecting their outside domains to similar domains in other octahedra would yield a lattice resembling the octahedral portion of a face-centered cubic lattice, but of lower symmetry.

Fig. 20. Components of a DX Octahedron Lattice

Figure 20. Components of a DX Octahedron Lattice. The drawing on the upper left contains an octahedron, three of whose edges contain a second domain. The second domain is indicated by a ball at either end and a ball in the middle, all connected by a linear stick. The three DX domains span a three dimensional space. The center of the octahedron is indicated by a small ball. The upper right contains a drawing of only the extra domains, but extended over two unit cells in each direction. The three drawings on the bottom show the complete octahedron twice, each time joined by a different one of the three domains.

Fig. 21. Trigonal View of a Lattice Made of DX Octahedra

Figure 21. Trigonal View of a Lattice Made of DX Octahedra. This is a view down the 3-fold axis of the lattice shown in Figure 20. Eight unit cells are shown. The 'impossible structure' interlacing of the extra domains is a consequence of the fact that the contents of only seven of the unit cells are visible in this projection.

Concluding Comments

DNA nanotechnology is a promising avenue to achieve the goals of nanotechnology in general. The specificity of DNA interactions combined with branched molecules represent a system whereby it is possible to gain large amounts of control over both linking and branching topology. Two features of the system remain to be developed. One of these, discussed above, entails the construction of periodic matter, including the attachment of guests and pendent molecules. As noted above, this will give us a rational means for determining macromolecular structure by generating crystals for x-ray diffraction experiments [28], as well as allowing us to direct the assembly of arrays of other molecules besides DNA [29]. Among the targets for x-ray diffraction experiments, one must include complex knots and catenanes: We can demonstrate the synthesis of the simplest members of these classes by gel electrophoresis, but more complex topological figures require direct physical observation. Winfree has proposed using DX arrays in DNA-based computing [40]. That approach, too, requires the ability to build periodic backbones, although the bases would differ from unit cell to unit cell.

The other goal for DNA nanotechnology does not require periodic matter. This is the use of DNA structural transitions to drive nanomechanical devices. Two transitions have been mentioned prominently, branch migration and the B-Z transition. It is known that applying torque to a cruciform can lead to the extrusion or intrusion of a cruciform [41]. A synthetic branched junction with two opposite arms linked can relocate its branch point in response to positive supercoiling induced by ethidium [42]. The experimental system used to demonstrate this level of control is illustrated in Figure 22. This molecule represents the very first step in using DNA structural transitions to achieve a nanomechanical result. We are also exploring the use of the B-Z transition in nanomechanical devices.

Fig. 22. Control of Branch Migration

Figure 22. An Experiment Demonstrating Control of Branch Migration. The features of the molecule used in this experiment are illustrated at the top left of the drawing. It is a circular duplex molecule containing a tetramobile branched junction. The four mobile nucleotides on each strand are drawn to be extruded from the main circle. There are 262 nucleotides in the circle to the base of the extruded junction, 4 mobile pairs, 12 immobile pairs above the mobile section, and a tetrathymidine loop in each strand, for a total of 298 nucleotides in each strand. The molecule is constructed from three segments, a duplex consisting of strands L1 and L2, a duplex consisting of strands R1 and R2, and the tetramobile junction, consisting of strands JT and JB. The divisions between the segments are indicated by vertical lines, except that the 5' ends of JT and JB are indicated by starbursts, indicating the 5' radioactive phosphate labels that are attached individually for analysis (never in pairs). These starburst sites are the scission points of EcoR V and Sca I restriction nucleases. The immobile junction contains Pst I and Stu I restriction sites, which are indicated. The experiment is done by positively supercoiling the circle in order to relocate the branch point by means of branch migration; this is shown in the transition to the upper right of the drawing. The positive supercoiling is achieved by adding ethidium. The molecule is then cleaved by the junction resolvase, endo VII (lower right). Following endo VII cleavage, the molecule is restricted (center bottom), and the points of scission are analyzed on a sequencing gel (lower left).

The ideas behind DNA nanotechnology have been around since 1980 [43]. However, the realities of experimental practice have slowed their realization. No experiment works in the laboratory as readily as it works on paper: One must obtain proper conditions, refine designs and determine experimental windows through the tedious and often expensive process of trial and error. Many of these are in place now for the goals outlined above. The past few years have witnessed increasing interest in the field. Mirkin, Letsinger and their colleagues [44] have attached DNA molecules to colloidal gold, with the goal of assembling nanoparticles into macroscopic materials, and more recently for diagnostic purposes [45]. Alivisatos, Schultz, and their colleagues [46] have used DNA to organize nanocrystals of gold. Niemeyer et al. have used DNA specificity to generate protein arrays [47]. Shi and Bergstrom have attached DNA single strands to rigid organic linkers; they have shown that they can form cycles of various sizes with these molecules [48]. It is to be hoped that this marked increase in experimental activity will lead to the achievement of its key goals within the near future.

Acknowledgments

This research has been supported by grant GM-29554 from the National Institute of General Medical Sciences, grant N00014-89-J-3078 from the Office of Naval Research, and grant NSF-CCR-97-25021 from the National Science Foundation.

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