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Proc Natl Acad Sci U S A. 2018 Jul 17;115(29):E6690-E6696. doi: 10.1073/pnas.1722681115. Epub 2018 Jul 3.

Universal folding pathways of polyhedron nets.

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Chemical Engineering Department, University of Michigan, Ann Arbor, MI 48109.
Applied Physics Program, University of Michigan, Ann Arbor, MI 48109.
Chemical Engineering Department, University of Michigan, Ann Arbor, MI 48109;
Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109.
Biointerfaces Institute, University of Michigan, Ann Arbor, MI 48109.


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 [Formula: see text] below which nets fold via nonnative contacts (bonds must break before the net can fold completely) and above which nets fold via native contacts (newly formed bonds are also present in the folded structure). Folding above [Formula: see text] shows a universal balance between reduction of entropy via the elimination of internal degrees of freedom when bonds are formed and gain in potential energy via local, cooperative edge binding. Exploiting this universality, we devised a numerical method to efficiently compute all high-temperature folding pathways for any net, allowing us to predict, among the combined 86,760 nets for the remaining Platonic solids, those with highest folding propensity. Our results provide a general heuristic for the design of 2D objects to stochastically fold into target 3D geometries and suggest a mechanism by which geometry and folding propensity are related above [Formula: see text], where native bonds dominate folding.


cooperativity; folding; origami; polyhedra nets

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