skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Universal folding pathways of polyhedron nets

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 T x 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 T x 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 T x , where native bonds dominate folding.

Authors:
; ;
Publication Date:
Research Org.:
Northwestern Univ., Evanston, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1458802
Alternate Identifier(s):
OSTI ID: 1540284
Grant/Contract Number:  
#DE-SC0000989; SC0000989
Resource Type:
Published Article
Journal Name:
Proceedings of the National Academy of Sciences of the United States of America
Additional Journal Information:
Journal Name: Proceedings of the National Academy of Sciences of the United States of America Journal Volume: 115 Journal Issue: 29; Journal ID: ISSN 0027-8424
Publisher:
Proceedings of the National Academy of Sciences
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; Science & Technology; Other Topics; folding; origami; polyhedra nets; cooperativity

Citation Formats

Dodd, Paul M., Damasceno, Pablo F., and Glotzer, Sharon C. Universal folding pathways of polyhedron nets. United States: N. p., 2018. Web. doi:10.1073/pnas.1722681115.
Dodd, Paul M., Damasceno, Pablo F., & Glotzer, Sharon C. Universal folding pathways of polyhedron nets. United States. doi:10.1073/pnas.1722681115.
Dodd, Paul M., Damasceno, Pablo F., and Glotzer, Sharon C. Tue . "Universal folding pathways of polyhedron nets". United States. doi:10.1073/pnas.1722681115.
@article{osti_1458802,
title = {Universal folding pathways of polyhedron nets},
author = {Dodd, Paul M. and Damasceno, Pablo F. and Glotzer, Sharon C.},
abstractNote = {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 T x 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 T x 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 T x , where native bonds dominate folding.},
doi = {10.1073/pnas.1722681115},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
number = 29,
volume = 115,
place = {United States},
year = {2018},
month = {7}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1073/pnas.1722681115

Citation Metrics:
Cited by: 2 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

General purpose molecular dynamics simulations fully implemented on graphics processing units
journal, May 2008

  • Anderson, Joshua A.; Lorenz, Chris D.; Travesset, A.
  • Journal of Computational Physics, Vol. 227, Issue 10
  • DOI: 10.1016/j.jcp.2008.01.047

A kirigami approach to engineering elasticity in nanocomposites through patterned defects
journal, June 2015

  • Shyu, Terry C.; Damasceno, Pablo F.; Dodd, Paul M.
  • Nature Materials, Vol. 14, Issue 8
  • DOI: 10.1038/nmat4327

The Free-Energy Landscape of Clusters of Attractive Hard Spheres
journal, January 2010


Forward flux sampling for rare event simulations
journal, October 2009

  • Allen, Rosalind J.; Valeriani, Chantal; Rein ten Wolde, Pieter
  • Journal of Physics: Condensed Matter, Vol. 21, Issue 46
  • DOI: 10.1088/0953-8984/21/46/463102

Protein folded states are kinetic hubs
journal, June 2010

  • Bowman, G. R.; Pande, V. S.
  • Proceedings of the National Academy of Sciences, Vol. 107, Issue 24
  • DOI: 10.1073/pnas.1003962107

Cell Origami: Self-Folding of Three-Dimensional Cell-Laden Microstructures Driven by Cell Traction Force
journal, December 2012


Pebble game algorithms and sparse graphs
journal, April 2008


Describing Protein Folding Kinetics by Molecular Dynamics Simulations. 1. Theory
journal, May 2004

  • Swope, William C.; Pitera, Jed W.; Suits, Frank
  • The Journal of Physical Chemistry B, Vol. 108, Issue 21
  • DOI: 10.1021/jp037421y

Self-folding polymeric containers for encapsulation and delivery of drugs
journal, November 2012


A Roadmap for the Assembly of Polyhedral Particles
journal, July 2012


Protein folding mechanisms and the multidimensional folding funnel
journal, August 1998


Complex wireframe DNA origami nanostructures with multi-arm junction vertices
journal, July 2015


Building Polyhedra by Self-Assembly: Theory and Experiment
journal, October 2014

  • Kaplan, Ryan; Klobušický, Joseph; Pandey, Shivendra
  • Artificial Life, Vol. 20, Issue 4
  • DOI: 10.1162/ARTL_a_00144

Finding the Optimal Nets for Self-Folding Kirigami
journal, May 2018


Nonlinear Machine Learning of Patchy Colloid Self-Assembly Pathways and Mechanisms
journal, April 2014

  • Long, Andrew W.; Ferguson, Andrew L.
  • The Journal of Physical Chemistry B, Vol. 118, Issue 15
  • DOI: 10.1021/jp500350b

The protein folding problem: when will it be solved?
journal, June 2007

  • Dill, Ken A.; Ozkan, S. Banu; Weikl, Thomas R.
  • Current Opinion in Structural Biology, Vol. 17, Issue 3
  • DOI: 10.1016/j.sbi.2007.06.001

Strong scaling of general-purpose molecular dynamics simulations on GPUs
journal, July 2015

  • Glaser, Jens; Nguyen, Trung Dac; Anderson, Joshua A.
  • Computer Physics Communications, Vol. 192
  • DOI: 10.1016/j.cpc.2015.02.028

How does symmetry impact the flexibility of proteins?
journal, February 2014

  • Schulze, Bernd; Sljoka, Adnan; Whiteley, Walter
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 372, Issue 2008
  • DOI: 10.1098/rsta.2012.0041

Convex polytopes with convex nets
journal, November 1975

  • Shephard, G. C.
  • Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 78, Issue 3
  • DOI: 10.1017/S0305004100051860

Native contacts determine protein folding mechanisms in atomistic simulations
journal, October 2013

  • Best, R. B.; Hummer, G.; Eaton, W. A.
  • Proceedings of the National Academy of Sciences, Vol. 110, Issue 44
  • DOI: 10.1073/pnas.1311599110

Mesophase behaviour of polyhedral particles
journal, February 2011

  • Agarwal, Umang; Escobedo, Fernando A.
  • Nature Materials, Vol. 10, Issue 3
  • DOI: 10.1038/nmat2959

Using Markov state models to study self-assembly
journal, June 2014

  • Perkett, Matthew R.; Hagan, Michael F.
  • The Journal of Chemical Physics, Vol. 140, Issue 21
  • DOI: 10.1063/1.4878494

A method for building self-folding machines
journal, August 2014


Transition Path Theory for Markov Jump Processes
journal, January 2009

  • Metzner, Philipp; Schütte, Christof; Vanden-Eijnden, Eric
  • Multiscale Modeling & Simulation, Vol. 7, Issue 3
  • DOI: 10.1137/070699500

Excess chemical potential of dilute solutions of spherical polyelectrolytes
journal, April 1974

  • Phillies, George D. J.
  • The Journal of Chemical Physics, Vol. 60, Issue 7
  • DOI: 10.1063/1.1681434

Transition-Path Theory and Path-Finding Algorithms for the Study of Rare Events
journal, March 2010


Linking rigid bodies symmetrically
journal, November 2014


Constructing the equilibrium ensemble of folding pathways from short off-equilibrium simulations
journal, November 2009

  • Noé, Frank; Schütte, Christof; Vanden-Eijnden, Eric
  • Proceedings of the National Academy of Sciences, Vol. 106, Issue 45
  • DOI: 10.1073/pnas.0905466106

Predictive Self-Assembly of Polyhedra into Complex Structures
journal, July 2012


On the shortest spanning subtree of a graph and the traveling salesman problem
journal, January 1956


Controlled Origami Folding of Hydrogel Bilayers with Sustained Reversibility for Robust Microcarriers
journal, November 2011

  • Shim, Tae Soup; Kim, Shin-Hyun; Heo, Chul-Joon
  • Angewandte Chemie, Vol. 124, Issue 6
  • DOI: 10.1002/ange.201106723

Graphene kirigami
journal, July 2015

  • Blees, Melina K.; Barnard, Arthur W.; Rose, Peter A.
  • Nature, Vol. 524, Issue 7564
  • DOI: 10.1038/nature14588

Cooperativity in protein-folding kinetics.
journal, March 1993

  • Dill, K. A.; Fiebig, K. M.; Chan, H. S.
  • Proceedings of the National Academy of Sciences, Vol. 90, Issue 5
  • DOI: 10.1073/pnas.90.5.1942

Anisotropy of building blocks and their assembly into complex structures
journal, August 2007

  • Glotzer, Sharon C.; Solomon, Michael J.
  • Nature Materials, Vol. 6, Issue 8, p. 557-562
  • DOI: 10.1038/nmat1949

Rigidity of multi-graphs. I. Linking rigid bodies in n-space
journal, February 1984


Algorithmic design of self-folding polyhedra
journal, December 2011

  • Pandey, S.; Ewing, M.; Kunas, A.
  • Proceedings of the National Academy of Sciences, Vol. 108, Issue 50
  • DOI: 10.1073/pnas.1110857108

Rigid body constraints realized in massively-parallel molecular dynamics on graphics processing units
journal, November 2011

  • Nguyen, Trung Dac; Phillips, Carolyn L.; Anderson, Joshua A.
  • Computer Physics Communications, Vol. 182, Issue 11
  • DOI: 10.1016/j.cpc.2011.06.005

Algorithmic lattice kirigami: A route to pluripotent materials
journal, May 2015

  • Sussman, Daniel M.; Cho, Yigil; Castle, Toen
  • Proceedings of the National Academy of Sciences, Vol. 112, Issue 24
  • DOI: 10.1073/pnas.1506048112

Contact order and ab initio protein structure prediction: CONTACT ORDER AND AB INITIO PROTEIN STRUCTURE PREDICTION
journal, August 2002

  • Bonneau, Richard; Ruczinski, Ingo; Tsai, Jerry
  • Protein Science, Vol. 11, Issue 8
  • DOI: 10.1110/ps.3790102

The number of nets of the regular convex polytopes in dimension ⩽ 4
journal, May 1998


Self-folding thin-film materials: From nanopolyhedra to graphene origami
journal, September 2012


Compactness Determines the Success of Cube and Octahedron Self-Assembly
journal, February 2009