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# Universal folding pathways of polyhedron nets

Contributed by Sharon C. Glotzer, June 6, 2018 (sent for review January 17, 2018; reviewed by Nuno Araujo and Andrew L. Ferguson)

## Significance

What makes an object successful at thermal folding? Protein scientists study how sequence affects the pathways by which chained amino acids fold and the structures into which they fold. Here we investigate the inverse problem: Starting with a 3D object as a polyhedron we ask, which ones, among the many choices of 2D unfoldings, are able to fold most consistently? We find that these “nets” follow a universal balance between entropy loss and potential energy gain, allowing us to explain why some of their geometrical attributes (such as compactness) represent a good predictor for the folding propensity of a given shape. Our results can be used to guide the stochastic folding of nanoscale objects into drug-delivery devices and thermally folded robots.

## Abstract

Low-dimensional objects such as molecular strands, ladders, and sheets have intrinsic features that affect their propensity to fold into 3D objects. Understanding this relationship remains a challenge for de novo design of functional structures. Using molecular dynamics simulations, we investigate the refolding of the 24 possible 2D unfoldings (“nets”) of the three simplest Platonic shapes and demonstrate that attributes of a net’s topology—net compactness and leaves on the cutting graph—correlate with thermodynamic folding propensity. To explain these correlations we exhaustively enumerate the pathways followed by nets during folding and identify a crossover temperature

In the 16th century, the Dutch artist Albrecht Dürer investigated which 2D cuts of nonoverlapping, edge-joined polygons could be folded into Platonic and Archimedean polyhedra. Dürer cuts were later called “nets” but, for a long time, the interest around them was mostly restricted to the field of mathematics (1⇓–3). A newer concept, self-folding origami, adds a modern twist to the ancient art of paper folding. By providing a mechanism to achieve complex 3D geometries from low-dimensional objects—without the need for manipulation of the constituent parts—self-folding brings the concepts pioneered by Dürer to the forefront of many research fields, from medicine (4) to robotics (5).

Several recent works have leveraged physical forces to achieve controllable folding of 3D objects including light (6), pH (7), capillary forces (8), cellular traction (9), and thermal expansion (10). Other works have investigated the relationship between geometric attributes of the object being folded and its propensity for successful folding. In the macroscopic folding of kirigami sheets—origami-like structures containing cuts and creases—the effect of different cut patterns on the material’s stress–strain behavior has been elucidated (11⇓–13) and the “inverse design problem” of finding cuts leading to the folding of a particular target structure has been solved (14). For submillimeter-sized capsules, formed via nonstochastic folding of nets into polyhedra, it has been suggested that nets fold with higher yield when they are compact (8, 15, 16), but the reason for this correlation remains unclear. In natural systems, the canonical example of self-folding occurs for proteins, where a string of amino acids navigate, thermodynamically, from a denatured (unfolded) state to a natured (folded) one. Even after many decades of study, however, a universal relationship between molecular sequence and folded state—which could provide crucial insight into the causes and potential treatments of many diseases—remains out of reach (11⇓–13).

In this work, we study the thermodynamic foldability of 2D nets for all five Platonic solids. Despite being the simplest and most symmetric 3D polytopes, the family of Platonic shapes suffices to demonstrate the rapid increase in design space as shapes become more complex: A tetrahedron has 2 possible net representations, cubes and octahedra each have 11 nets, and dodecahedra and icosahedra have, each, 43,380 distinct edge unfoldings. Here we are interested in the thermodynamic self-folding of these nets. Our goal is to understand how topology affects yield in the stochastic folding of 3D objects. The advantage is threefold. First, by using a collection of sheets folding into the same target shape, we isolate the geometric attributes responsible for high-yield folding. Second, the model allows exhaustive computation of the pathways followed by the nets during folding, elucidating how some nets achieve high yield. Third, by studying increasingly more complex objects—from tetrahedra to icosahedra—we can use the folding mechanisms quantified in the simplest objects to predict, and potentially validate, their occurrence in the more complex shapes.

Beginning with a target Platonic shape (Fig. 1), we construct a graph whose vertices and edges correspond to those in the polyhedron. This mapping of the shape to a graph facilitates the exhaustive search of all distinct nets by allowing spanning tree enumeration (3). A set of prechosen edges (the cutting tree) is then cut, in a process called edge unfolding, to create a single, contiguous, and flat 2D sheet of nonoverlapping faces: a net. For the Platonic shapes, whose nets are enumerated (17), this can be repeated exhaustively until all distinct nets are discovered (see *Materials and Methods* for more details). We note that for other shapes, while the process of computing all nets might become computationally prohibitive, it has been recently demonstrated that a subset of interest for these nets can be computed algorithmically (18). We list all 86,784 nets for the Platonic solids in a database (19).

Each net is modeled as a sheet of rigid polygons connected to adjacent polygons via harmonic springs. The polygons are composed of rigidly connected spheres and the influence of thermal fluctuations on a single net, suspended in implicit solvent, is modeled via Langevin molecular dynamics (more details in *Materials and Methods*). We assign nonspecific, short-ranged attractive (sticky) interactions between all edges not joined by springs of a net so that the polyhedron formed from folding is also the ground-state configuration. This does not guarantee, however, the uniqueness of the ground state and, as we will see, other 3D foldings can arise. Unless explicitly noted otherwise, by “folded state” we refer to the original polyhedron.

As in proteins and other biomolecules, the nonspecificity of the interactions between edges of the nets allows for both native and nonnative contacts. As a consequence, when the system temperature is rapidly decreased (quenched), kinetic traps are possible and net misfolds are observed. This possibility for kinetic traps raises the following question: Among all nets generated by unfolding a polyhedron, which of them show the highest propensity to refold into the original polyhedron?

## More Compact, “Leafy” Nets Fold More Reliably

To identify the nets able to fold reliably into their polyhedron of origin we performed hundreds of cooling simulations for each net, using both a fast and a slow cooling protocol (see *Materials and Methods* section for more details). The two distinct nets for the tetrahedron, hereafter referred to as the triangular net and the linear net (Fig. 2*A*), showed remarkably different folding propensities for the fast cooling protocol: Of 125 simulations, all triangular nets folded into the target tetrahedron while only 54% of the linear nets succeeded—the other 46% of the experiments resulted in misfolded configurations. In general the slower cooling rate simulations yielded a higher folding probability for each net. This is expected as the net has more time to find the global minimum. Similar simulations for each of the 11 nets of both the cube and the octahedron revealed even larger differences: For the cube nets (Fig. 2*B*), only 3 of the nets showed greater than 50% success in folding, and only at slow cooling rates; for the octahedron (Fig. 2*C*), none of the nets achieve higher than 50% success rate for either cooling rate.

The misfolded configurations for the tetrahedron and cube nets were incomplete 3D geometries (showing, for instance, faces collapsed on top of each other or bonds incompatible with the formation of the respective target shape). In contrast, octahedron nets often folded into another 3D shape: a concave, boat-like conformation with the same number of edge–edge contacts as the octahedron: in other words, a degenerate ground state. This competing structure, which is less symmetric than the octahedron, has higher rotational entropy resulting in a lower free energy than the octahedron (see *SI Appendix*, Fig. S1*A* for folding probabilities for the boat conformation). A competition between similar degenerate structures was reported for finite clusters of six attractive spherical colloids (20), where symmetry breaking leads to the formation of the same boat-like conformation.

Fig. 2 shows that, despite having the same ground-state energy, not all nets of a polyhedron are equivalent. In general, we observe that the nets that fold most reliably are the most compact and have the highest number of leaves on its cutting graph (green solid circles in nets in Fig. 2). A net is said to be more compact if it has a large number of leaves (the vertices with degree one on the cutting tree) and a small diameter (the longest shortest path between any two faces on the face graph). Exact values for the leaves and diameter are shown in *SI Appendix*, Table S1. Most strikingly, even nets differing only by the location of a single face can have folding probabilities reduced from 99% to 17%. What causes one shape to fold nearly perfectly every time while a slightly different one fails to do so almost as frequently? And why do net “leafiness” and “compactness” correlate with folding yield?

## High-Temperature Folding Happens via Native Contacts

To answer why small differences in net topology can have a large impact on the net’s folding propensity, we used Markov-state models (MSM) (21⇓–23) to compute the pathways through which nets fold into their folded state. Since quench rate was observed to affect the folding propensity of the nets, we run constant temperature simulations of each net while computing the rate of transitions between two states (flux). A representative folding network is shown in Fig. 3*A* (see *SI Appendix*, Fig. S2 for other example networks). Arrows represent observed transitions between different states and each arrow has a thickness proportional to the probability flux of the transition being observed (see *Materials and Methods* for more details). To simplify, we show only the most visited pathways (i.e., those whose combined flux accounts for at least 50% of the total reactive flux between unfolded and folded states). Intermediate configurations can achieve the folded state via the formation of native (green) or nonnative (red) contacts. If a pathway includes native contacts only, every newly formed bond is compatible with the final polyhedron and error correction is not needed. When the folded state is achieved via incompatible bonds, we observe that these nonnative contacts can sometimes have an active role by bringing otherwise far-away native contacts closer together, facilitating folding. In other cases the nonactive contacts contribute only passively and folding occurs sequentially after the misfold occurs until it reaches a point where the nonnative contact must break for folding to continue. In either case, nonnative contacts must eventually break before a folded configuration can be achieved.

The total flux connecting unfolded and folded states (Fig. 3*B*) first increases with temperature and then decreases to zero at high temperature. At low *T* the folding flux is low because the states are mostly trapped into a few configurations; i.e., the slow kinetics inhibit bond breaking. As the temperature increases, bonds can now break and the folding/unfolding process occurs at a higher rate. At intermediate T, a maximum in reactive flux is observed, meaning that there is a temperature at which there is a maximum number of expected transitions from unfolded to folded state per unit time τ. The representative network shown in Fig. 3*A* was calculated at that peak temperature. Finally, at high T the unfolded state is preferred and again the flux vanishes. These trends are also true for the other nets studied (*SI Appendix*, Fig. S3). If we separate the flux into those following native and nonnative contacts, we see (Fig. 3*C*) that the folding pathways at high temperature mostly follow the formation of native contacts while the behavior inverts for low temperatures, and mostly nonnative contacts are observed. There were only two nets where a crossover temperature was not observed. The triangular tetrahedral net does not exhibit a crossover temperature since it has no traps. The best-folding cubic net also does not have a crossover temperature in the range of temperatures we study; while traps exist, the fraction of native contacts decreases and the temperature decreases but never falls below 50% (*SI Appendix*, Fig. S3). This T dependency is also observed in colloids, where the assembly of an icosahedron is monomeric at high temperatures, but, at low temperature, particles first aggregate into large clusters (not necessarily compatible with icosahedral symmetry) and those later rearrange into the ground-state structure (24).

To gain an understanding of the mechanisms underlying the observation that native contacts are favored at high temperatures during folding, we calculated the number of degrees of freedom associated with each intermediate.

## Nets Follow a Universal Balance Between Entropy and Enthalpy

The fact that more compact nets and those with many leaves generally fold with higher yield suggests that nets might fold locally, in a manner that reduces the fewest degrees of freedom, thereby maximizing the conformational entropy along the folding pathways. To test whether this trade-off between maximizing degrees of freedom and forming native contacts occurs at high T, we calculate the number N of internal degrees of freedom and the number Q of native contacts as a net folds. We do so for all 24 nets of the tetrahedron, cube, and octahedron. Fig. 4 shows that, remarkably, all nets follow a folding pathway that achieves a narrow balance between reduction of degrees of freedom and gain of potential energy. In practical terms, high-temperature folding happens locally such that, at each step of the process, the system strives to maximize its conformational entropy. From this observation, we hypothesize the following mechanism for the folding of general nets at high temperature. Folding should primarily happen (*i*) via nearby (local) connections, favoring compact nets with many leaves, and (*ii*) along one of the optimal trade-off paths, favoring nets with high degeneracy in the number of such optimal paths. Fig. 4 shows that, remarkably, all nets follow a folding pathway that achieves a narrow balance between reduction of degrees of freedom and gain of potential energy. In practical terms, high-temperature folding happens locally such that, at each step of the process, the system strives to maximize its conformational entropy. Importantly, this maximization does not necessarily occur at low temperatures as when intermediate nonnative contacts occur—and stages of the folding are found to lie far away from the narrow balance explained above (*SI Appendix*, Fig. S4).

Using this hypothesis we devised an algorithm to generate high-temperature pathways for the 86,760 nets of the dodecahedron and icosahedron, without the need for a full MSM calculation. We do so by first enumerating candidate bonds that can be made if they are next to each other on the net or in the intermediate and then by selecting ones that have the largest number of degrees of freedom (see *Materials and Methods* for more details). Fig. 5 shows the combined example pathways followed by all 11 nets for the cube (Fig. 5*A*) and for the octahedron (Fig. 5*B*), illustrating the pathways that maximize degrees of freedom and using only local, native contacts to fold. We then used these principles to find, among the 86,760 nets, those with high and low folding yields (see *SI Appendix*, Figs. S5 and S6 for specific nets). The corresponding folding propensities are plotted in Fig. 6. As expected, polyhedra with many leaves and small diameter show higher propensity for correct folding and those nets also have many high-T pathways to the ground state. The correlation between the folding propensity and the leaves may have analogues in other systems as well. The number of leaves is a measure of the amount of local connections that are required to fold from the unfolded state. In the protein-folding literature the contact order is a measure of how far away specific native contacts are on the amino acid sequence and it has been shown that low contact order inversely correlates to folding rate (25).

Overall, our observations suggest that folding propensity of a net decreases as the number of faces increases. For instance, while the 4-sided tetrahedron folds nearly perfectly, the 20-sided icosahedron is unable to successfully fold. One exception to this trend is the dodecahedron, which folds with higher probability than the octahedron. While the reason for this exception remains elusive, there are two factors that may play a role. The first factor is the number of degrees of freedom retained by the faces sharing a leaf vertex when the leaf edges form a bond. For instance, octahedron nets can make a bond about the leaf vertex, but due to the unique symmetry the loop, retain 1 df (i.e., the resulting intermediate is not rigid), so the intermediate may still enter a trapped state. In the case of the icosahedron, after folding about leaf vertices, the loop retains 2 df. In the case of the other three shapes (i.e., tetrahedron, cube, dodecahedron), folding about leaf vertices renders the loop rigid. This implies that further constraining the net by increasing the rigidity throughout the folding process is important to sufficiently funnel the net’s energy landscape and may boost the folding probability for many nets. The second factor is the complexity that arises in trapped states. There are “tetrahedral motifs” on many octahedral nets, and these motifs can fold into full or partial tetrahedra, as seen in the boat conformation mentioned above (see *SI Appendix*, Fig. S1*A* for more examples). For icosahedral nets, there are both tetrahedral and octahedral motifs, and the diversity of the trapped states is further increased for these nets. This is in sharp contrast to the nets of the other three shapes, for which trapped states typically occur when one face folds onto another face.

## Discussion and Conclusion

The observed preference for native contact pathways at high temperature is not unique to polyhedron nets. Several small proteins have been observed in simulation to fold via native-only contacts when close to their melting temperature (26). At low T (or high hydrophobicity) the pathways shift to a hydrophobic collapse, in which nonnative contacts form followed by further rearrangements leading to the native state (27). Similarly, colloids assemble via monomeric pathways at high temperature, forming bonds that are compatible with the overall structure in an equivalent process to native contact formation (24). Finally, systems of colloidal sticky spheres prefer to form the same concave, boat-like conformation that we observe for octahedron nets (20).

Our simple model therefore draws connections between the macroscopic irreversible folding of polyhedra (8), assembly of patchy (24) and colloidal (20) particles, and the folding of amino acids (26⇓–28). The identified trade-off between entropy and enthalpy that dictates high-temperature folding provides guiding principles for the assembly of 3D complex geometries from potentially simpler-to-fabricate 2D nets. We demonstrated the judicious pathway engineering via the selection of nets with certain characteristics. We found that nets at high temperature fold through pathways that maximize the internal degrees of freedom, regardless of their propensity to fold, and the more compact nets fold with higher propensity. The compactness measures correlated with the number of pathways connecting the unfolded and folded states, offering some understanding on why these measures work well. In addition to giving insights into the thermodynamics of folding in naturally occurring systems, our results could also provide a route for the fabrication of anisotropic Brownian shells, paving the way for the self-assembly of complex crystals from nanoshells and colloidal shells (29⇓⇓–32) capable of encapsulating cargo. We expect these results to impact future experiments on folding of graphene sheets (13), graphene oxide layers (12), or DNA–origami polyhedral nets (33).

## Materials and Methods

### Enumeration of Polyhedral Nets.

We create nets from polyhedra via a process known as edge unfolding. In edge unfolding, one cuts along a set of prechosen edges, called the cutting tree, of a polyhedron (e.g., a cube) to create a single, contiguous flat 2D sheet of connected (square) faces: a net. We enumerate all of the nets of each polyhedron by generating random weights for the edges of the skeleton-1 graph of the polyhedron on the interval [0, 1]. The minimal spanning tree was found using Kruskal’s algorithm (34). We then converted the spanning tree to a net and added it to the database if it did not already exist. We ran this loop for many iterations until we found all of the nets for each shape.

### Langevin Dynamics and Molecular Dynamics Simulation.

Langevin dynamics are used to model the folding dynamics for each net using HOOMD-Blue (35⇓⇓–38). Each face of the net is approximated by a union of spheres acting as a rigid body, with an edge length of 10 spheres. The spheres are arranged in a hexagonal lattice for triangular faces, a square lattice for square faces, and a hexagonal approximate for the pentagonal faces. For each simulation the drag coefficient, γ, was set to the inverse of the number of spheres used to create a facet. The spheres in the center of the face interact via a Weeks–Chandler–Andersen (WCA) potential shown in blue in Fig. 1, while the spheres on the free edges of the net interact via a Lennard–Jones potential; both potentials used ϵ and σ values of 1.0. The rigid facets are tethered together using harmonic springs along the edges, using a spring constant of 800 and equilibrium distance of 1.0.

### Quantifying Yield.

To quantify the folding yield we ran 125 simulations starting from a high temperature and quenched the temperature to near zero (*T* ≤ *T*_{m} + 2.5, where *SI Appendix*, Table S1.

### MSMs and Folding Pathway Calculations.

MSMs (21) have been used to study protein folding (22, 23) and virus capsid assembly (39) and can provide a detailed view into the dynamics and thermodynamics of the folding landscape. In contrast to other methods to compute assembly pathways, which require monotonic order parameters (40) or hierarchical computation of the partition function (41), MSMs require each simulation snapshot to be classified as a discrete state, and the number of transitions between each state is recorded in a matrix. The dihedral angles completely specify the configuration of a net and are therefore a good set of collective variables. We break the “up”/“down” folding degeneracy by keeping track of the dihedral angle on the interval

To build the MSM we ran 125 independent simulations at constant volume and temperature for *SI Appendix*, Fig. S7). We used transition path theory (23, 43, 44) to determine the reactive flux,

### Folding Parameters.

The number of native contacts, Q, was calculated by counting the number of edges that were bonded to the correct corresponding edge according to the criteria described above. The nonnative contacts were calculated similarly. The diameter is the graph diameter of the face graph of the net. In general the number of degrees of freedom can be difficult to calculate because it can be difficult to deduce which constraints are redundant in the net. In general one can use the pebble game (45); however, if the linkage is not generic or has point group symmetries, the pebble game can underestimate the number of degrees of freedom (46, 47). First, we applied the pebble game to each intermediate, and then for closed-loop motifs we applied a closed-loop formula to determine the number of degrees of freedom (48); finally, intermediates with high degrees of symmetry were checked by hand, since the pebble game is known to underestimate these cases (46, 47).

### Enumerating High-Temperature Pathways.

An exhaustive search was performed to enumerate high-T pathways. Two principles were assumed to be important for the folding pathways at high temperature: local bonds and maximizing number of degrees of freedom. We initialize the algorithm by adding the unfolded state to the queue and creating an empty graph that will contain the pathway information. For each intermediate on the queue, a set of candidate bonds was calculated by finding edges on the intermediate that still needed to be bonded that also had a topological distance of one (local). This intermediate was then added to a queue for further processing, and a link between the current state and the candidate state is made in the graph. Finally, the pathways are taken from the graph and sorted lexicographically by the sequence of degrees of freedom of each intermediate along the pathway. The pathways that have the largest number of degrees of freedom are then returned.

## Acknowledgments

This work was supported in part by the National Science Foundation, Emerging Frontiers in Research and Innovation (EFRI) Award EFRI-1240264 (to P.M.D.) and as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0000989. P.F.D. acknowledges support from the University of Michigan Rackham Predoctoral Fellowship Program. Computational work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant ACI-1053575, XSEDE Award DMR 140129, and was also supported in part through computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: sglotzer{at}umich.edu.

Author contributions: P.M.D., P.F.D., and S.C.G. designed research; P.M.D. implemented the methods and performed numerical simulations; P.M.D., P.F.D., and S.C.G. analyzed data; and P.M.D., P.F.D., and S.C.G. wrote the paper.

Reviewers: N.A., University of Lisbon; and A.L.F., University of Illinois at Urbana–Champaign.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1722681115/-/DCSupplemental.

Published under the PNAS license.

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