Origami Life

Abstract

Regardless of the scale it is built at, the geometric principles of origami make it an architecture both constrained by the act of folding and imbued with dynamics by folding. From space stations to molecules, origami solutions are inspiring a new type of design that considers its deployment as an integral part of the structure. In this essay, I will explain some surprising similarities between the ancient art of folding paper origami and the folding of RNA at the nanoscale.

The invention of folded paper is likely to have arrived very shortly after the invention of paper. The simple act of folding the sheet, something so seemingly obvious you might not think to give it a name, began to organically grow into the elegant art form we today call ‘Origami’. Inspired by over a thousand years of tradition and innovation in paper folding, modern scientists and engineers have gained a new appreciation for origami design [1]—as folding solutions in paper seem to translate upward to the scale of buildings [2] and downward to the scale of molecules [3]. An origami begins by developing a pattern of creases in a sheet of paper; the folds allow the flat sheet of paper to be reconfigured into different geometric forms. The basic act of folding is one of the transformations: The mere placement of creases on a flat sheet allows for nearly any imaginable three-dimensional animal or plant or abstract form to be created—an act almost as magical as life springing out from a seed.

Folding and packing are the central aspects of paper origami, but are the rules of origami universal to all folded things? Much like origami, folding and packing are both extraordinarily important for the stabilization of structural RNA molecules, which play essential roles in regulation, chromosome maintenance and protein biosynthesis [4]. Are there lessons to be learned from origami that can help us to better navigate the problems of biopolymer folding in a tactile way?

Similar to how living animals develop from the confines of a single cell that divides, an origami model begins as creases within the bounds of a sheet of paper. Origami seems to be an ideal medium for mimicking the shapes and forms found in nature. Origami expert Jun Maekawa points out that in many traditional origami models, the colored top surface becomes the outside and the bottom of the sheet becomes the inside, and so as a consequence, the origami has the same topology as a blastula [5]. Modern origami techniques are now even able to mimic the stripes of zebras [6] and spots on beetles [7] by using both sides of the paper strategically. Nature perfected folding before humans invented origami, and examples can be observed all over nature; in the unfurling petals of flowers, the unfolding of insect wings, or the way that some plants dry out and coil up to spread their pollen or seeds. In all of these cases, the pattern of folds provides major evolutionary advantages to the organism. The ability to contract and expand (or deploy into a larger shape) is the mechanism of motion for all living things, even at a molecular level. From the way that our genes are densely packed into a fractal-like network of coiled coils, to how protein expression is dynamically turned on and off just when needed, deployable structures are part of the mechanism of life. The art of origami is indeed a beautiful representation of this concept.

Some of the earliest innovations in the art of paper folding originate from Japan. Traditionally folded from very high-quality paper, it was mostly used for ceremonial purposes, at weddings or as decorations on gifts [8]. It was not until around the early 1900s in Japan that books formalizing the art were published, and the name ‘Origami’ became associated with paper folding. The Japanese paper crane and variants, along with a handful of traditional models from other cultures, such as the paper balloon and pinwheel fold, represented the total knowledge of paper forms at the time. Then, in the 1950s, Akira Yoshizawa entered the scene and both revolutionized and popularized origami when he published ‘Atarashii Origami Geijutsu’ (New Origami Art), introducing many innovative new origami models and techniques for shaping origami, and perhaps even more importantly, standardized the diagrams used to depict the folding process [9]. The seeds were planted, and the ingenuity of the art form was accelerated from there. As the complexity of creasing patterns increased, the intricacy of the resulting forms began to gain origami international recognition as a new medium for sculpture. By the 1980s, there were thousands of published origami designs.

The formalization of axioms describing the geometry of origami [10] and contributions from innovators in origami design, such as Maekawa’s system for designing complex origami folds [5], inspired the creation of computer algorithms [11] that are able to automate much of the design of crease patterns for origami. The most notable of these algorithms, TreeMaker [12], written by Robert Lang in 1996, was the first to enable the design of origami with an arbitrary number of flaps of any length, which become the arms and legs of the origami (see Fig. 1a). TreeMaker led to a ‘Cambrian explosion’ of new complexity in origami folding among origami designers as they raced to generate astonishingly intricate models with ever more flaps with which to sculpt extra details like additional legs, claws, feathers and scales.

Fig. 1

Modularity in origami. a an origami ‘molecule’ fold based on the isosceles right triangle, one of the simplest folds that defines a flap. The lines show the position of folds, and the blue circle slice represents the area of the paper that contributes to the flap (right). b the four traditional bases of Japanese origami: the kite base, the fish base, the bird base and the frog base. Each of the bases tiles a different number of units of the isosceles triangle molecule shown in a, either 2, 4, 8 or 16. The blue areas show the part of the paper that will make a flap. c the folded form of each of the origami bases with 1, 2, 4 or 5 flaps, respectively. d an example origami folded from each base. Designs with greater number of flaps achieve much higher compaction in the final model. e an example of a modular folding pattern. The waterbomb base can be tiled such that each row is offset to the middle of its neighbors above and below. Figures a–d adapted from [13]

After many contributions from mathematicians, physicists and engineers [1], origami has become a highly multidisciplinary area of science. There are endless practical applications for folding, as origami can be found in many places where size and deployability are important, such as heart stents, airbags, or folding solar panels. Here, I will share a story about folding, how origami is a physical analogy for biological folding landscapes, and why multistability is both important and inherent in folded systems.

1 Origami Molecules

Origami is an art form that continually pushes the limits of our understanding of the mechanics of folding. As artists strive to pack more detail and information into the creases confined to the bounds of a sheet of paper, designs can become so intricate that they take many hours to fold by hand. Origami models are often folded in two main stages: Initially an origami base is made, a large interconnected fold using the entire paper that creates a number of flaps (Fig. 1b, c). Next, the flaps that were created are sculpted into details such as legs or wings and other features (Fig. 1d). The base functions as a scaffold to organize flaps, and the flaps are used to create all the details that define the model.

Surprisingly, despite how diverse the variety of origami has become, most origami features the repeated usage of key folds, which are called ‘molecules’ in the Japanese tradition [13]. Molecule folds, of which there are many varieties, turn out to be especially useful because of their modularity within origami designs. One of the most recurrent molecule folds of origami, shown in Fig. 1a, is an isosceles triangle that is bisected by a crease at 22.5°, defining a single flap. Two copies of the molecule fold can be fit into a square (Fig. 1b, leftmost) to produce a kite base, the simplest of the four classic bases of origami: kite, fish, bird and frog. The incredible practicality of this fold is that, due to its shape, it can be scaled successively smaller to generate all of the traditional Japanese origami base folds (Fig. 1b), and this pattern can even be extended to larger designs [13].

Starting from the same molecule fold (Fig. 1a), each of the traditional origami bases produces different numbers of flaps (Fig. 1c), and the area of paper used for each flap is highlighted in blue on the crease pattern (Fig. 1b). The folded base can be further sculpted into familiar origami forms by adding additional detail folds to the flaps (Fig. 1d). The resulting origamis become smaller with respect to the paper as more flaps and details are added. To compensate for this fact, origami artists often use very large paper up to 50 cm square or larger, to produce the most complex models [14]. In the process of exploring how to subdivide the paper to maximize the number and size of flaps, artists discovered and documented many new rules for design: For example, common origami base folds, such as the waterbomb fold (Fig. 1e, left), can be tiled into larger and more complex folds (Fig. 1e, right). The concept of fold modularity in origami was pushed to new limits when several innovative origami designers at the time began devising new origami by combining molecule folds in new patterns [13]. The modularity of folding origami allows for common folds to be merged in surprising ways. John Montroll used this concept to elegantly create a bird base with a fifth extra flap [15], and Jun Maekawa created the elongated features of a crocodile [5] by merging two different traditional base folds.

Around the same time that the field of origami was having a breakthrough with respect to modular design principles, scientists around the world were just beginning to experiment with modular strands of DNA produced by new solid-phase synthesis technology—initiating the fields of DNA and RNA nanotechnology. The lab of Nadrian Seeman was investigating ways to program DNA strands to assemble into stable three-dimensional configurations, such as a cube [16]; and in this pursuit, they designed a new fold of DNA that would become the ‘origami molecule’ for nanotechnology—the double crossover junction [17]. Just as early progress in origami design was made by creating variations elaborated from one of a few different origami base folds, the field of nanotechnology grew by building larger and more complex structures using the double crossover as a structural strut [18] and by folding genomes into repeating arrays of double crossovers that we call scaffolded DNA origami [3].

In parallel with developments in DNA nanotechnology, a theory of structural modularity for RNA molecules was being developed [19]. Westhof, Masquida and Jaeger hypothesized that RNA is modular and that new structures can be composed by rearranging and merging fragments of existing structures. This theory would lay the groundwork for developing a modular design method for constructing RNA structures [20]. RNA is the working component behind many biologically important molecular machines, responsible for protein synthesis and information processing in all living cells. Because the RNA machines of life have a modular construction that we can learn to use, there are many possibilities for the rational design of RNA by recombining structural modules.

Early work exploring the modularity of RNA tertiary interactions demonstrated that modular design could be applied to create programmable nanoscale assembly units of RNA [21]. Modular RNA structures self-assemble into defined shapes guided by the incorporation of different aptamers, junctions and programmable connectors embedded within their sequence. The full versatility of structural modularity for RNA initially proposed by Jaeger and colleagues [19, 20] was demonstrated by creating a building system of modular RNA assembly units that link up to form defined assemblies and repeating lattices (Fig. 2a) [22]. By merging different structural modules corresponding to bends, junctions and connectors, arbitrary and highly-detailed designs such as nano-hearts were produced (Fig. 2b). RNA architectures can be rationally mapped to a single-stranded path, encoded into a gene, and then expressed as self-folded shapes [22]. Although in RNA the folds are encoded into sequences of nucleotides, rather than formed by creases in origami paper, it is the same property of modularity of folds that makes it possible to fold paper into origami animals and compose RNA structures that self-assemble (or fold) into nanoscale shapes.

Fig. 2

Modularity of RNA origami. a programmable RNA lattices formed from square and triangle RNA units. b self-assembling RNA hearts from multiple units (top) or produced cotranscriptionally from a single strand (bottom). c the ‘origami molecule’ fold of RNA origami, a kissing loop interaction stabilizes the formation of a double-crossover section (inset and yellow box). On the edge of the structure are terminal hairpins, analogs to ‘flaps’ of origami, that can be functionalized with a variety of different RNA structural motifs such as programmable kissing loops, protein binding sites and fluorescent light-up aptamers such as Spinach and Mango. d larger and more complex RNA structures can be built by merging together copies of the fold in a. RNA origami structures range from 300 to 2300nts in scale. e functionalization of an RNA origami scaffold with binding sites for proteins fused to CFP and YFP. Adjacent binding sites bring the CFP and YFP proteins close enough to produce FRET. Figures a–b adapted from reference [22]. Figures c–e adapted from reference [25]

Drawing inspiration from the incredible structures of DNA origami [3] built using the double-crossover motif [17], I developed an RNA version of the motif that has a single-stranded topology [23]. In DNA origami, numerous short staple-strands help a long scaffold strand to fold into a compact form. Now, in RNA origami, programmable kissing loops (KLs) define one of the edges of the double-crossover motif and consequently enables designs based on this pattern to be produced cotranscriptionally (Fig. 2c). KL interactions span the space of a normal helix, but function to both connect and coaxially align distant helices. The single-stranded crossover fold, which I will call a KL crossover, is the equivalent of an origami ‘molecule’ fold for RNA and forms a compact modular unit that provides an underlying structure for numerous helices that can function as ‘flaps’ (Fig. 2c, d). Each unit of the KL crossover fold creates a geometric closure via the formation of a kissing loop [24], adding a growing element of stability to the origami, especially when cooperatively forming many KLs in alignment. As a scaffold material, RNA origami designs comprise a core of multiple KLs that are flanked by hairpins (Fig. 2c, e). These helix ‘flaps’ can be functionalized with a toolbox of programmable connectors, protein binding sites, or fluorescent light-up aptamers such as Spinach or Mango (Fig. 2c, inset).

RNA origami structures containing two, four, six or more copies of the KL crossover unit can be constructed because of the structural modularity, increasing design complexity through simple repetition [25] (Fig. 2d) similar to the molecule folds of paper origami (Fig. 1b, e). Analogous to the way that flaps are developed with detail folds in an origami model (Fig. 1d), the free arms of the RNA origami structure can be embedded with RNA motifs that ultimately decide the functionality of the structure. For demonstration, RNA origami was produced containing binding sites for two fluorescent protein complexes that can interact to produce FRET, a type of resonant energy transfer that can only occur when fluorophores are within a few nanometers of each other (Fig. 2e). Validating that origami scaffolds can colocalize and position proteins with precision, scaffolds created with adjacent binding sites produced a much brighter FRET response than scaffolds designed with far-apart binding sites or no binding sites at all [25] (Fig. 2e). In a paper origami, the folds of the base are all interconnected, while the folds of details are local and can be adjusted without compromising the base form. Similarly, RNA origami architecture uses a repeating modular pattern of interconnected local folds to form a stable core structure, out from which project helical arms can be sculpted with details from natural biological folds to guide protein binding, position aptamers, or other RNA active sites with precision.

2 Origami Design Algorithms

The culmination of years of exploration in the modularity of origami folds by numerous experts led to a mathematical formalization of paper origami and folding [10], systematic new approaches to origami design [5] and framework for creating arbitrarily complex folds [11]. These methods introduced a new school of thought to the origami world, designs became abstracted as tree-like stick figure diagrams, and the placement of origami flaps became distilled into a computational optimization problem. In particular, the TreeMaker [12] algorithm was very influential in the origami world because it enabled a new level of control over the number and relative size of every flap in a design. With this software, users can computationally generate creasing patterns for a wide variety of new origami.

Origami design using TreeMaker begins by studying the subject to represent in origami, in this example, a lizard (Fig. 3a). First, the lizard is abstracted as a weighted tree graph (Fig. 3b); each leaf on the tree will be folded into a separate flap, and the weights reflect the relative lengths of those body segments. Circles with radii weighted by the tree are fit, so their centers are within the bounds of the paper and then scaled so that their perimeters all touch but do not overlap (Fig. 3c)—each circle represents where a flap will be created. The distance between a center of a circle and that of another matches the weight of the path between the corresponding leaves on the edge-weighted tree. In this example, those weights happen to all be length one, and so, for example, the distance between a front leg and a back leg is three (Fig. 3c). The blue ‘river’ area of the design that does not contain any circles functions like a flap that is connected at both ends and will be what the body of the design is folded from (Fig. 3c–e). Next, the algorithm fills in the polygons with molecule fold patterns (Fig. 3d). The ‘rabbit ear’ molecule is a general fold solution for a triangular space, and more complex molecule patterns exist that allow any polygon to be filled with creases [26]. During the folding compaction, the creases will function as hinges, and the polygons of paper between the creases are called ‘facets’. As the fold is collapsed, all of the facets move in concert (Fig. 3e), and the cross-section of the fully collapsed base fold has the same shape as the tree (Fig. 3b). Once the origami is folded, the addition of detail folds [13] and other well-established shaping techniques, such as wet folding [9], can be applied to further refine the origami so that it resembles a lizard (Fig. 3e).

Fig. 3

Origami design with TreeMaker. a a lizard, as inspiration for design. b a tree diagram abstraction of the lizard, where each branch has a length of one. c to represent the tree in the bounds of the paper, each branch of the tree becomes abstracted as a circular area, and the body becomes the blue ‘river’ section, arranged to maximize the packing. Triangles on the graph connecting the nodes have lengths corresponding to the path length along the tree. d each polygon of the main crease pattern can be filled using an origami ‘molecule’ fold. Here, a general example for triangles is shown where creases bisecting each angle fold the triangle into three flaps. Each flap is colored according to the node it is assigned to in c. e sequence depicting the collapsing of the crease pattern into folded form with the same cross-section as the tree graph. With sculpting, the base fold can then be made to look like a lizard. Figure adapted from reference [12]

While for paper origami, algorithms simplify the work to produce complicated models by automating the design of folds starting from a tree diagram; in RNA, algorithms are used instead to create and optimize a sequence that will fold into the desired tree structure. An RNA can be thought of as a kind of one-dimensional origami that is deployed by the act of transcription; the leaves and branches of the tree structure and its folding pathway are encoded into the sequence of the RNA strand. Be it creating a pattern of paper creases that define a shape [27] or a pattern of nucleotides that define a fold [28], design is difficult—and the time to compute a design increases rapidly with the complexity of the design. Once the design is completed, it can even take an origami master considerable time and patience to fold and collapse a complex crease pattern. In comparison, an RNA origami transcript that is designed well will fold all by itself! However, the key phrase is ‘designed well’, and because of the complexity of the inverse folding problem, design optimization by computer algorithms is absolutely essential.

Recently, I developed the RNA origami design software ROAD [25], which enables researchers to model, optimize and encode large and complex RNA folds into a single-stranded sequence (Fig. 2d), allowing RNA origami nanostructures that are kilobases in scale to be produced. ROAD, in a similar manner to TreeMaker, is an iterative optimization algorithm that attempts to design origami structures based on an inputted tree. However, rather than design crease patterns, ROAD designs a sequence of nucleotides that encodes folds in the RNA strand. The problem of inverse folding by sequence design is approached by guessing an initial sequence and then repeatedly revising that design to improve its score based on a number of heuristic tests. This approach relies on repeatedly evaluating guesses in an energy model that simulates RNA folding and benefits from fast algorithms such as ViennaRNA for reliably computing RNA folds [29]. However, pseudoknots, the interactions that form between distal single-stranded regions of a structure, are notoriously difficult to compute—spelling a potential problem for designs that require numerous KL connectors.

While many software already exist to design RNA, two features unique to ROAD make it ideal for producing RNA origami designs: First, it is able to rapidly assign origami KL sequences by systematically cross-checking all loops against each other, whereas other software that are able to design pseudoknots do not scale to such large designs. Second, it designs sequences with the constraints of cotranscriptional synthesis as a main consideration, avoiding specific sequences and patterns known to interfere with transcription or the experimental workflow. While early modular RNA designs were produced by hand-aligning structural modules to model RNA particles, for example, RNA squares created with exchangeable motifs for the corners [30] or tilings of small square and triangular assemblies (Fig. 2a) [22], the size and complexity of designs were ultimately very limited in scale compared to what can now be achieved with the computer-aided modeling and design offered by ROAD (Fig. 2d) [25].

The development of paper origami and DNA/RNA origami has both been greatly accelerated by computer-aided design algorithms. However, just as the crease patterns generated by TreeMaker do not provide step-by-step folding instructions and do not guarantee that the compacted design will be an elegant one, RNA origami designs, even with computer-optimized sequences, still may not fold correctly if the order and staging of folding events are not planned well. Ambiguity of fold definitions, the topology of the strand/paper and the order of folding events are all factors determining the shape of the folding landscape for origami structures.

3 Origami Folding Pathways

Multistability occurs whenever a system has two or more stable equilibrium states or as a consequence of a folding pathway that gets stuck in non-equilibrium states. The presence of competing stable states is a common consequence of complex and dynamic systems, and origami is no exception. When folding a paper origami, certain combinations of mountain and valley folds produce bistability. A classic example of this is the waterbomb fold (Fig. 4a), which can transition between a triangular and square forms by inversion. The square form of the waterbomb is also commonly called the ‘Preliminary Fold’ [15] and is a starting fold for many origami models.

Fig. 4

Bistability in origami and RNA. a the waterbomb fold can collapse into two entirely different structures and is the most common example of bistability in origami. b a simple bistable RNA structure; it can alternate between either two short or one long hairpin. The energy barrier to transition between the structures can be as high as the unfolded state, indicated by arrow. Figure adapted from [31]. c the adenine riboswitch is a classic example of bistability. A color-coded arc representation of the competing secondary structures (left) shows how the terminator hairpin has bistability with stems P1 and P3. d schematic of adenosine sensing, depicting a window during transcription where the switch can activate based on ligand binding. If the aptamer is not stabilized in time, then the terminator stem forms and prevents the production of the rest of the RNA strand. Figures c–d adapted from reference [32]

The central vertex of the waterbomb fold is a ‘node’ that moves to actuate the inversion of the fold. As a node of the waterbomb is inverted, the fold becomes unstable at the transition point between the two states when the paper is fully flattened. This is because paper has non-zero thickness; creasing the paper sets a new preferred angle in the fibers of the paper along the fold lines, leading to the paper resisting being fully flattened. In the case of the waterbomb fold inversion, elastic deformation of the crease away from its preferred angle during the transition stores energy as the paper ‘pops’ from one fold to the other as you force the transition [33]. In other cases, such as for concentrically pleated folds, energy can be stored in the bending of non-triangular facets as the paper balances trying to be perfectly flat along the facets with the position and bend of the creases [34]. Bistability in origami is caused by competing forces between the creases wanting to bend and the facets of paper between the creases resisting bending.

In RNA, an analogous bistable fold to the waterbomb would be two short hairpins that can transition into a single longer hairpin (Fig. 4b). Just as bistable folds can be found all over origami, bistable sequences are naturally prevalent in RNA and furthermore are so simple to design that a bistable sequence can be produced for any pair of two structures [31]. The tendency for RNA sequences to have multiple folded states of similar energy can present a real challenge for producing well-folded RNA structures. And likewise just as for origami, multistability and also topology are both major factors in how RNA sequences fold. Multistability is a property of sequences that is often used in important ways by functional RNA molecules. Because RNA is synthesized directionally from 5’ to 3’, kinetic control of folding in RNA can be used to direct folding down a specific folding pathway. For example, the adenine riboswitch (Fig. 4c) is the classic example of bistability and can be controlled by tipping the stability balance with ligand binding at an early point during the transcription of the sequence [32]. Similarly, the more recently characterized ZTP riboswitch navigates its folding landscape in a complicated manner, but its behavior is ultimately determined by ligand binding during a narrow window of time early in the transcription [35]. In a linear folding pathway, the ability to freely transition between bistable alternatives occurs during the narrow window of time when the strand is actively folding.

In paper origami, the crease patterns will frequently have multistable elements with many bistable node points. In order for such a structure to be folded correctly, all of the nodes need to point the correct direction at the start of the collapse (Fig. 3e). Attempting an origami collapse with even a single misaligned node will lead to a conformationally trapped configuration that cannot flatten any further (Fig. 5a). This type of conformational trap is similar to the way that biopolymers can become kinetically trapped during their folding. Two basic properties in common between biopolymers and origami allow the problem of conformational trapping to arise: First, the paper or molecular strand cannot be stretched or compressed, only folded. Second, the paper or molecular strand cannot intersect itself in the folded form nor at any intermediate folding state [36].

Fig. 5

Origami conformational traps. a a hypothetical energy landscape for a paper crane. Similar to the waterbomb fold, the bird base has a bistable landscape, although only one of the two folds can collapse fully and be made into a crane. Starting at the unfolded state, the central node can be displaced either up or down, leading to either the misfolded or collapsed state, respectively. The final folding of the model into a crane is represented by a large energy barrier. b a hypothetical energy landscape for the hairpin ribozyme junction, showing the magnesium-induced stabilization of the folded and collapsed states. Magnesium binding (red dots) overcomes electrostatic repulsion and allows the RNA to condense into tightly packed structure upon rearrangement. Alternative helix stacking order can produce partially stabilized structures that cannot fully compact. An interaction between the green and blue helices drives the final stabilization of the collapsed structure into the folded structure. c the order of folds can lead to bifurcation, for example, adjacent valley folds can lead to different stacking orders. Additionally, if either segment A or C is longer than B, then one of the two possible stacking orders becomes conformationally blocked. Figure c adapted from reference [36]

To illustrate how conformational trapping can occur, consider the crease pattern for the paper crane base, one of the simplest variants of the waterbomb fold that also has a trapped state (Fig. 5a). The energetic landscape of a waterbomb fold has been measured using mechanical models [33], and for the hypothetical crane fold I illustrate something similar but with one of the two states having a lower energy than the other (Fig. 5a, misfolded vs. collapsed). The highest energy point corresponds to the unfolded state of the paper, and this is because it takes energy to elastically deform the creases from their preferred angle. The central node point or ‘flap tip’ of the waterbomb fold pattern can be pushed either upward or downward, releasing energy as the structure begins to compact. If it starts to compact toward the misfolded state, it will eventually reach a point where the structure cannot compact any further (Fig. 5a, left). In order to reach the correct folded state from the misfolded state, the entire sheet must first go back through the unfolded state (Fig. 5a, right), a common obstacle in the compaction of complex crease patterns for paper origami [37]. Finally, to create the fully developed crane model from the collapsed crane base requires additional deformations, and is represented by a significant energy barrier (Fig. 5a, right).

Just like for paper origami, energy is required to unfold a misfolded RNA structure before refolding it into a correct state, and for RNA, the barrier to transition between alternative secondary structures is quite high due to how strong base pairs are (Fig. 4b). Thus, if an RNA falls into a kinetic trap while it is forming base pairs, it can take a long time to refold those misfolded regions depending on the height of the energy barrier between alternative structures. For this reason, it is important to consider not only the final folded structure, but also all of the possible intermediate structures, when evaluating if a particular RNA design is prone to be stuck in folding traps. Interestingly, it is possible at a theoretical level to intentionally design sequences with extremely long and winding low energy-barrier folding pathways [38]. This suggests that it might be possible to design RNAs that change shape over time or even perform complex computations by folding.

Folding via a sequential folding pathway, for example, as a riboswitch folds (Fig. 4d), can be advantageous for controlling the direction of bifurcated nodes. Traditional origami provides a set of precise folding steps to produce the paper crane (Fig. 5a) that entirely avoids any chance of misfolding. By contrast, crease patterns produced in computational origami can have hundreds of folds that would ideally need to collapse in a concerted manner; however, the origami folder is limited by their two hands. Computational patterns are not generated with any folding sequence and often contain large irreducible folds that are interlinked and need to be collapsed simultaneously, and so sequential folding may not even be possible for many patterns. It is an important point that collapsing folds can produce origami that could not otherwise be produced in a stepwise manner; this was first documented by Akira Yoshizawa in 1959 when he innovated his incredible Cicada origami model, which collapses in a single motion from a tessellation of eight copies of the bird base fold [9].

Even with all the proper creases in place, it can take a good measure of skill to collapse a computational origami pattern in this manner. Expert origami folders often approach this problem by pushing each flap tip vertex in the proper direction, bit by bit, over the entire pattern until it reaches a tipping point and is able to compact [36]. Recent advances in the computer simulation of paper origami are able to estimate facet bending strain and model an idealized simultaneous collapse of all the creases in a design in a continuous motion [39]. The concerted collapse of a crease pattern echoes the annealing process that is often used to prepare DNA origami [3], whereby DNA strands are heated until they melt apart and then slowly cooled to let all interactions condense simultaneously. While DNA origami is typically folded from hundreds of short staple strands that work together to force a long scaffold strand to fold into a three-dimensional structure, there are also DNA and RNA origami structures that can collapse from a single long strand by annealing over a long temperature ramp [23, 40]. By contrast, the more biologically relevant cotranscriptional preparation of RNA origami is comparable to the stepwise folding instructions of traditional origami. Interestingly, the folding space for heat-annealed RNAs and cotranscriptionally folded RNAs does appear to be quite different [23], with annealing having the ability to produce some structures that kinetic folding was not able to, much like what was observed to be the case for paper origami folding by Yoshizawa [9].

In addition to multistability in base pairing, such as is utilized to channel the folding in riboswitches, RNA structures also frequently encounter multistability in the collapse of multi-helix junctions. This is because the problem of folding is not just specifying the helices that fold, but also how the helices should be oriented with respect to one another. The helix packing stage of RNA folding can take substantially longer than the initial folding [41], especially for larger structures with multiple junctions that need to rearrange. This type of structural rearrangement is depicted in Fig. 5b, showing the folding of the hairpin ribozyme [42]. In the hairpin ribozyme, the blue and orange helices can coaxially stack different ways, but only one conformation is further stabilized by an interaction with the green helix (Fig. 5b). Compared to multistability in base pairing (Fig. 4b), the energy barrier to move between alternate packing conformations is much lower (Fig. 5b), meaning that RNAs can explore this structural space more freely during folding. During the folding of RNA, magnesium counterions bridge and neutralize the negative charges between helix backbones, triggering a collapse into more compact structures [43]. Stabilization by binding metal ions has the effect of deepening the free energy curve for the fully folded structure over misfolded structures, in effect channeling folds to the correct structure given enough time [44].

Deepening the energy well for the desired fold can be used to rationally design RNA structures for which junctions fold rapidly and stably into the correct stacking conformation. The strategy of forming geometric closures through long-range interactions is widespread in biology [24] and is an effective way to stabilize multi-helix junctions. The hairpin ribozyme uses a long-range interaction to specify the antiparallel orientation of the junction [45], and likewise, other arrangements can be programmed based on the context of tertiary interactions. For example, embedding a loop–receptor interaction into two stems of a four-way junction can strongly favor a parallel-helix stacking orientation that aligns the loop and receptor [46]. In RNA origami structures, the same strategy is used, and both parallel and antiparallel arrangements can be specified by choosing the proper length of the KL-crossover [23].

In paper origami, the order in which the folds are made can lead to different stacking orders of the paper layers (Fig. 5b). For a given crease pattern, determining a valid stacking order is computationally involved because of how interdependent the folds are [36]. Consider the simple placement of just two adjacent valley folds (Fig. 5c), if either segment A or C is longer than B, then one of the two possible stacking orders becomes conformationally blocked. The context dependence of folding makes it difficult to predict how facets in a crease pattern may interact, as what happens at the crease between segments B and C depends on how segment A was folded. In larger origami designs, these dependencies can be far more complex, often linking folds on opposite ends of a sheet [36].

Not surprisingly, context dependence is a major factor in RNA folding as well; just as the folding of paper can block later creases from folding, the folding of strands into pseudoknots during transcription can create folding barriers [47]. Locally, the energy of each base pair is dependent on its neighbors, and globally, the final compaction of helices into three-dimensional structure is driven by the stacking of helices at junctions and by the cooperativity of numerous weaker packing interactions [48]. The order in which the folds occur within an RNA origami can have a large influence on the outcome. As an RNA strand folds cotranscriptionally, the rate of folding into helices is many times faster than the rate of strand synthesis; long-range interactions such as KLs form comparatively slowly, giving helices and junctions time to stabilize before they lock into place. This hierarchy in rates makes it possible to design arbitrarily mazelike strand paths that arrange perfectly when KLs link up. Since each KL contributes roughly ~10 kcal/mol of binding energy in typical folding conditions, the total energetic contribution from many KLs can be substantial.

RNA origami is held together by numerous KLs working in concert, and the order in which the loops pair up in an RNA origami can lead to the formation of structural barriers. ROAD design software attempts to predict structural barriers by analyzing the position of KLs relative to the rest of the structure to see if any unpaired helices are nested within [25]. It is however up to the designer to refine their folding path based on the feedback. A recently developed program designed to optimize tree structures for cotranscriptional folding can help with this process by proposing structural variations that may have fewer barriers [49].

To illustrate the challenge of designing for sequential folding, consider a complex RNA tree (Fig. 6a) with branch lengths designed to fill the space of a rectangle (Fig. 6b). If we trace a path outlining the tree, it will make a representation of its secondary structure (Fig. 6c). Depending on where the 5’ and 3’ ends are placed on this circular path, different strand paths can be generated that represent the same fold—here, two examples are illustrated (Path1 and Path2, Fig. 6c). In the first example (Path1, Fig. 6d), the polymerase produces a long and winding structure that folds into a rectangular tile, condensing mostly at the end as the last KLs are put into place. At every intermediate position of the folding, there are no issues with the strand intersecting itself. In a second folding example (Path2, Fig. 6e), the location of the 5’ end is moved such that several KL pairs are made very early in the fold. When the KLs lock into place in this example, it creates several looped-out portions of single strand that still need to pair (Fig. 6e, orange strands). Since the RNA forms a helical turn every 11 base pairs, when the complementary portion of any of the longer orange sections is produced, it is going to encounter a structural barrier because the strands cannot intersect. As a result, some KLs will have to unpair before the red-colored portion can fold correctly (Fig. 6e, red strands). Cotranscriptional folding experiments comparing these two designs found that the Path1 design resulted in a considerably higher yield of correctly folded products compared to the Path2 design (Fig. 6de, right) [25]. While both Path1 and Path2 produced folded products, cotranscriptional folding via Path1 produced a much more homogeneous product at a higher yield. Furthermore, only Path1 produced TEM data of high enough quality to create an ab initio reconstruction of the RNA origami (Fig. 6d, right).

Fig. 6

RNA folding pathways. a a tree structure describing the secondary structure of an RNA origami. b the same tree, redrawn according to interactions between leaves on the tree. c the tree redrawn as a circular strand path, and aligning interacting helices. Two candidate positions (Path1 and Path2) place the 5’-end starting point at different points in the strand, arrow indicates strand direction. d Path1, beginning at the right edge, has a long circuitous path. An illustration of hypothetical folding intermediates is shown. A model of the folded structure is shown on the right, along with an ab initio structure produced by analyzing TEM micrographs. e Path2, starting in the middle of the design, forms a stable core structure with many long loops (highlighted in orange). As the structure folds, it can reach a midpoint where the continued folding is blocked (shown in red) because the growing chain needs to wrap around a trapped region (orange) to pair with it. Yields of Path2 were lower than for Path1, and the TEM data for Path2 were not high quality enough for an ab inito reconstruction. Figure adapted from reference [25]

Misfolding appears to be a natural consequence of producing more complex folds. As more layers and more interactions between folds are added to a structure, the chance to create bifurcations during folding increases. Just as the wrong order of folds in a paper origami can lead to a conformation that blocks further folding (Fig. 5), the wrong order of synthesis through a single-stranded design can also lead to the formation of roadblocks to folding (Fig. 6e). The directional synthesis of RNA is one way to solve the problem of folding bifurcations, both with respect to the order of condensing helices, but also when alternative structures with equivalent or similar energies are present (Fig. 4b). Indeed, in the classic example of bistability, riboswitches take advantage of the slow and directional synthesis to control their own folding landscapes (Fig. 4d) [32]. Likewise, natural RNAs have evolved clever ways to cope with conformational multistability, as the hairpin ribozyme uses long-range interactions to lock the correct conformation of a junction into place [42]. Lastly, having a strand path that avoids getting tangled while it folds can result in higher yields and more homogeneous folded RNAs (Fig. 6d) [25].

4 Folded Origins

The word ‘Origami’ derives from two Japanese words: ‘ori’ meaning to fold and ‘kami’ meaning paper. Rather than describing the final folded product, the name origami refers to the active process by which it is produced. United by the idea of growth from folds, paper folding and biomolecular folding are both ultimately expressions of life. The two types of folding are governed by two limitations that are at the core of the concept of folding in general: The first is the property of non-intersection (injectivity) and the second is the property of non-stretching and non-compression of the material (isometry) [36]. Out of these two properties, almost all of the other features of folding emerge as a consequence.

Paper folding provides a tactile and intuitive way to understand the concept of biological folding landscapes, especially conceptually difficult ideas like folding traps. Paper is a versatile medium, it can be cut and glued, it has memory in the form of creases, and it can be dampened and reshaped [13]. It turns out that nucleic acids have many of the same properties: although ribozymes and ligases do the cutting and gluing, base pairing and magnesium-induced stabilization provide the memory and glue—achieving effectively a similar result. In this way, DNA and RNA origamis become a natural extension of an art form that has thrived for over a thousand years.

In both paper origami and RNA origami, we encounter numerous types of bifurcations in folding pathways and solve them in similar ways. From multistable helices in riboswitches to the many conformers of helical junctions, the problem of folding always requires consideration of its deployment when building structure. In the final vision of any architecture, there are often many unseen layers of logistics and organization that are critical to enable the construction. Perhaps, this is the most elegant aspect of it all, that there is a hidden beauty to folded architecture that is concealed by the dynamic process of its folding.

In spite of how well the analogy between origami and molecular folding works, there are also some interesting areas of contrast to mention: In origami, even ‘thick’ origami, the paper is usually much larger in dimensions compared to the thickness of the paper, so much so that the thickness of origami is easily idealized as zero-thickness in simulations. For RNA origami, a corresponding analogy might be to imagine the strand having nearly zero-thickness as well. However, when we zoom down to the scale of molecules, the thickness of the RNA helix and the length of the smallest helical features that can be designed are about the same, ~2 nm. As a result, RNA designs that have more structural features need to be made considerably larger than structures with fewer features—unlike paper which can be folded both more densely and smaller to achieve more features. Also, and particularly a contrast, when a biopolymer strand condenses into a misfolded state, it takes much longer to unfold than it does to misfold in the first place—with paper origami, it is quite the opposite and much more difficult to fold than it is to unfold!

The art of origami is an exploration of possibilities reached by folding and so is a reflection of life that is created through the act of folding. The RNA world hypothesis proposes that self-replicating biopolymers could have played a key role in the beginning of evolution [50]. In theory, life may have originated from the folding of a lifeless strand. Very recently, the Unrau lab made amazing progress toward producing a fabled ‘replicase’, a sequence of RNA that can copy itself [51]. The new ribozyme has a sense of ‘self’, using a bistable clamping mechanism that recognizes its own promoter sequence, demonstrating again how bistability is a fundamentally important property of living polymers. RNAs in nature navigate folding pathways to activate different functions in a context-dependent way. Today, we are only just beginning to unlock the secrets behind designing and folding RNA. An exciting new era for molecular structure that can deploy cotranscriptionally awaits!