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1091-6490121182024Apr30Proceedings of the National Academy of Sciences of the United States of AmericaProc Natl Acad Sci U S ALimits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra.e2315648121e2315648121e231564812110.1073/pnas.2315648121We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies-in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g., periodicity)-is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.DuqueCarlos MCM0000-0003-0567-4664Max Planck Institute of Molecular Cell Biology and Genetics, Dresden 01307, Germany.Center for Systems Biology Dresden, Dresden 01307, Germany.Department of Physics, University of Massachusetts, Amherst, MA 01003.HallDouglas MDM0000-0003-2829-6272Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003.TyukodiBotondBDepartment of Physics, Babes-Bolyai University, Cluj-Napoca 400084, Romania.Martin A. Fisher School of Physics, Brandeis University, Waltham, MA 02453.HaganMichael FMF0000-0002-9211-2434Martin A. Fisher School of Physics, Brandeis University, Waltham, MA 02453.SantangeloChristian DCDDepartment of Physics, University of Massachusetts, Amherst, MA 01003.Department of Physics, Syracuse University, Syracuse, NY 13210.GrasonGregory MGM0000-0001-5479-9370Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003.engDMR-2011846National Science Foundation (NSF)031L0160German Federal Ministry of Education and Research829010European Union's Horizon 2020 Research and Innovation Programme101026118European Union's Horizon 2020 research and innovation programmeDMR-2309635National Science Foundation (NSF)DMR-2217543National Science Foundation (NSF)Journal Article20240426
United StatesProc Natl Acad Sci U S A75058760027-8424IMaddressable assemblyprogrammable materialsself-assemblyself-closing assemblytriply periodic polyhedraCompeting interests statement:The authors declare no competing interest.20244261854202442618532024426125320241026ppublish38669182PMC1106705910.1073/pnas.2315648121Thompson J. M. T., Parker A. R., A vision for natural photonics. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 362, 2709–2720 (2004).15539366Prum R. O., Dufresne E. R., Quinn T., Waters K., Development of colour-producing β-keratin nanostructures in avian feather barbs. J. R. Soc. Interface 6, S253–S265 (2009). PMC270646919336345Neville A. C., Biology of Fibrous Composites (Cambridge University Press, Cambridge, 1993).Fratzl P., Cellulose and collagen: From fibres to tissues. Curr. Opin. Coll. Interface Sci. 8, 32–39 (2003).Perlmutter J. D., Hagan M. F., Mechanisms of virus assembly. Annu. Rev. Phys. Chem. 66, 217–239 (2015).PMC438237225532951Kerfeld C. A., Aussignargues C., Zarzycki J., Cai F., Sutter M., Bacterial microcompartments. Nat. Rev. Microbiol. 16, 277–290 (2018).PMC602285429503457Rother M., Nussbaumer M. G., Renggli K., Bruns N., Protein cages and synthetic polymers: A fruitful symbiosis for drug delivery applications, bionanotechnology and materials science. Chem. Soc. Rev. 45, 6213–6249 (2016).27426103Hagan M. F., Grason G. M., Equilibrium mechanisms of self-limiting assembly. Rev. Mod. Phys. 93, 025008 (2021).PMC888025935221384Oosawa F., Asakura S., Thermodynamics of the Polymerization of Protein (Academic Press, London, UK, 1975).Bouligand Y., Liquid crystals and biological morphogenesis: Ancient and new questions. C. R. Chim. 11, 281–296 (2008).Michielsen K., Stavenga D., Gyroid cuticular structures in butterfly wing scales: Biological photonic crystals. J. R. Soc. Interface 5, 85–94 (2008).PMC270920217567555Saranathan V., et al. , Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales. Proc. Natl. Acad. Sci. U.S.A. 107, 11676–11681 (2010).PMC290070820547870Wilts B. D., Michielsen K., Kuipers J., De Raedt H., Stavenga D. G., Brilliant camouflage: Photonic crystals in the diamond weevil, Entimus imperialis. Proc. R. Soc. B: Biol. Sci. 279, 2524–2530 (2012).PMC335069622378806Saranathan V., Narayanan S., Sandy A., Dufresne E. R., Prum R. O., Evolution of single gyroid photonic crystals in bird feathers. Proc. Natl. Acad. Sci. U.S.A. 118, e2101357118 (2021).PMC820185034074782Ke Y., Ong L. L., Shih W. M., Yin P., Three-dimensional structures self-assembled from DNA bricks. Science 338, 1177–1183 (2012).PMC384364723197527Jones M. R., Seeman N. C., Mirkin C. A., Programmable materials and the nature of the DNA bond. Science 347, 1260901 (2015).25700524Zeravcic Z., Manoharan V. N., Brenner M. P., Colloquium: Toward living matter with colloidal particles. Rev. Mod. Phys. 89, 031001 (2017).Douglas S. M., et al. , Self-assembly of DNA into nanoscale three-dimensional shapes. Nature 459, 414–418 (2009).PMC268846219458720Huang P. S., Boyken S. E., Baker D., The coming of age of de novo protein design. Nature 537, 320–327 (2016).27629638Glotzer S. C., Solomon M. J., Anisotropy of building blocks and their assembly into complex structures. Nat. Mater. 6, 557–562 (2007).17667968Sacanna S., Pine D. J., Shape-anisotropic colloids: Building blocks for complex assemblies. Curr. Opin. Coll. Interface Sci. 16, 96–105 (2011).Heuckel T., Hocky G. M., Sacanna S., Total synthesis of colloidal matter. Nat. Rev. Mater. 6, 1053–1069 (2021).Su Z., et al. , The role of architectural engineering in macromolecular self-assemblies via non-covalent interactions: A molecular LEGO approach. Progr. Polym. Sci. 103, 101230 (2020).Murugan A., Zou J., Brenner M. P., Undesired usage and the robust self-assembly of heterogeneous structures. Nat. Commun. 6, 6203 (2015).25669898Murugan A., Zeravcic Z., Brenner M. P., Leibler S., Multifarious assembly mixtures: Systems allowing retrieval of diverse stored structures. Proc. Natl. Acad. Sci. U.S.A. 112, 54–59 (2015).PMC429166425535383Jacobs W. M., Reinhardt A., Frenkel D., Rational design of self-assembly pathways for complex multicomponent structures. Proc. Natl. Acad. Sci. U.S.A. 112, 6313–6318 (2015).PMC444337025941388Jacobs W. M., Frenkel D., Self-assembly of structures with addressable complexity. J. Am. Chem. Soc. 138, 2457–2467 (2016).26862684Zeravcic Z., Manoharan V. N., Brenner M. P., Size limits of self-assembled colloidal structures made using specific interactions. Proc. Natl. Acad. Sci. U.S.A. 111, 15918–15923 (2014).PMC423455225349380Ke Y., et al. , DNA brick crystals with prescribed depths. Nat. Chem. 6, 994–1002 (2014).PMC423896425343605Ong L. L., et al. , Programmable self-assembly of three-dimensional nanostructures from 10,000 unique components. Nature 552, 72–77 (2017).PMC578643629219968Nykypanchuk D., Maye M. M., Van Der Lelie D., Gang O., DNA-guided crystallization of colloidal nanoparticles. Nature 451, 549–552 (2008).18235496Auyeung E., et al. , Synthetically programmable nanoparticle superlattices using a hollow three-dimensional spacer approach. Nat. Nanotechnol. 7, 24–28 (2012).22157725Tian Y., et al. , Ordered three-dimensional nanomaterials using DNA-prescribed and valence-controlled material voxels. Nat. Mater. 19, 789–796 (2020).31932669Rogers W. B., Crocker J. C., Direct measurements of DNA-mediated colloidal interactions and their quantitative modeling. Proc. Natl. Acad. Sci. U.S.A. 108, 15687–15692 (2011).PMC317911921896714Hensley A., Jacobs W. M., Rogers W. B., Self-assembly of photonic crystals by controlling the nucleation and growth of DNA-coated colloids. Proc. Natl. Acad. Sci. U.S.A. 119, e2114050118 (2022).PMC874076134949716Majewski P. W., et al. , Resilient three-dimensional ordered architectures assembled from nanoparticles by DNA. Sci. Adv. 7, eabf0617 (2021).PMC797842633741597Michelson A., et al. , Three-dimensional visualization of nanoparticle lattices and multimaterial frameworks. Science 376, 203–207 (2022).35389786Park S. H., Park H., Hur K., Lee S., Design of DNA origami diamond photonic crystals. ACS Appl. Biomater. 3, 747–756 (2020).35019418Gerling T., Wagenbauer K. F., Neuner A. M., Dietz H., Dynamic DNA devices and assemblies formed by shape-complementary, non-base pairing 3D components. Science 347, 1446–1452 (2015).25814577Rothemund P. W. K., et al. , Design and characterization of programmable DNA nanotubes. J. Am. Chem. Soc. 126, 16344–16352 (2004).15600335Benson E., et al. , DNA rendering of polyhedral meshes at the nanoscale. Nature 523, 441–444 (2015).26201596Sigl C., et al. , Programmable icosahedral shell system for virus trapping. Nat. Mater. 20, 1281–1289 (2021).PMC761160434127822Videbaek T. E., et al. , Tiling a tubule: How increasing complexity improves the yield of self-limited assembly. J. Phys.: Condens. Matter 34, 134003 (2022).PMC885704734983038Hayakawa D., et al. , Geometrically programmed self-limited assembly of tubules using DNA origami colloids. Proc. Natl. Acad. Sci. U.S.A. 119, e2207902119 (2022).PMC961814136252043Zandi R., Reguera D., Bruinsma R. F., Gelbart W. M., Rudnick J., Origin of icosahedral symmetry in viruses. Proc. Natl. Acad. Sci. U.S.A. 101, 15556–15560 (2004).PMC52484915486087Twarock R., Luque A., Structural puzzles in virology solved with an overarching icosahedral design principle. Nat. Commun. 10, 4414 (2019).PMC676502631562316Siber A., Icosadeltahedral geometry of geodesic domes, fullerenes and viruses: A tutorial on the t-number. Symmetry 12, 556 (2020).Johnson J. E., Speir J. A., Quasi-equivalent viruses: A paradigm for protein assemblies. J. Mol. Biol. 269, 665–675 (1997).9223631Caspar D. L. D., Klug A., Physical principles in the construction of regular viruses. Cold Spring Harb. Symp. Quant. Biol. 27, 1–24 (1962).14019094Bohlin J., Turberfield A. J., Louis A. A., Sulc P., Designing the self-assembly of arbitrary shapes using minimal complexity building blocks. ACS Nano 17, 5387–5398 (2023).36763807Pinto D. E. P., Sulc P., Sciortino F., Russo J., Design strategies for the self-assembly of polyhedral shells. Proc. Natl. Acad. Sci. U.S.A. 120, e2219458120 (2023).PMC1012001737040398Pedersen M. C., Hyde S. T., Polyhedra and packings from hyperbolic honeycombs. Proc. Natl. Acad. Sci. U.S.A. 115, 6905–6910 (2018).PMC614226429925600Tanaka H., Dotera T., Hyde S. T., Programmable self-assembly of nanoplates into bicontinuous nanostructures. ACS Nano 17, 15371–15378 (2023).PMC1044888537527198Lord E. A., Mackay A. L., Periodic minimal surfaces of cubic symmetry. Curr. Sci. 85, 346–362 (2003).Schoen A. H., Reflections concerning triply-periodic minimal surfaces. Interface Focus 2, 658–668 (2012).PMC343856824098851Schroeder G. E., Ramsden S. J., Christy A. G., Hyde S. T., Medial surfaces of hyperbolic structures. Eur. Phys. J. B: Condens. Matter 35, 551–564 (2003).Dotera T., Tanaka H., Takahashi Y., Hexagulation numbers: The magic numbers of equal spheres on triply periodic minimal surfaces. Struct. Chem. 28, 105–112 (2017).Sadoc J. F., Charvolin J., Infinite periodic minimal surfaces and their crystallography in the hyperbolic plane. Acta Crystallogr., Sect. A 45, 10–20 (1989).Ramsden S. J., Robins V., Hyde S. T., Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: Kaleidoscopic examples. Acta Crystallogr., Sect. A 65, 81–108 (2009).19225190Pedersen M. C., Hyde S. T., Hyperbolic crystallography of two-periodic surfaces and associated structures. Acta Crystallogr., Sect. A 73, 124–134 (2017).28248661Pedersen M. C., Hyde S. T., Ramsden S., Kirkensgaard J. J. K., Mapping hyperbolic order in curved materials. Soft Matter 19, 1586–1595 (2023).36749349C. T. Chantler, F. Boscherini, B. Bunker, Eds., International Tables for Crystallography: X-Ray Absorption Spectroscopy and Related Techniques (International Union of Crystallography, Chester, UK, ed. 1, 2020), vol. I.Tyukodi B., Mohajerani F., Hall D. M., Grason G. M., Hagan M. F., Thermodynamic size control in curvature-frustrated tubules: Self-limitation with open boundaries. ACS Nano 16, 9077–9085 (2022).PMC1036240335638478Rotskoff G. M., Geissler P. L., Robust nonequilibrium pathways to microcompartment assembly. Proc. Natl. Acad. Sci. U.S.A. 115, 6341–6346 (2018).PMC601682229866851Li S., Roy P., Travesset A., Zandi R., Why large icosahedral viruses need scaffolding proteins. Proc. Natl. Acad. Sci. U.S.A. 115, 10971–10976 (2018).PMC620549730301797Panahandeh S., et al. , How a virus circumvents energy barriers to form symmetric shells. ACS Nano 14, 3170–3180 (2020).32115940Fang H., Tyukodi B., Rogers W. B., Hagan M. F., Polymorphic self-assembly of helical tubules is kinetically controlled. Soft Matter 18, 6716–6728 (2022).PMC947259536039801Mohajerani F., et al. , Multiscale modeling of hepatitis b virus capsid assembly and its dimorphism. ACS Nano 16, 13845–13859 (2022), 10.1021/acsnano.2c02119.10.1021/acsnano.2c02119PMC1027325936054910Wagner J., Zandi R., The robust assembly of small symmetric nanoshells. Biophys. J. 109, 956 (2015).PMC456484326331253Li S., Zandi R., Travesset A., Grason G. M., Ground states of crystalline caps: Generalized jellium on curved space. Phys. Rev. Lett. 123, 145501 (2019).31702180Schwartz R., Shor P. W., Prevelige P. E., Berger B., Local rules simulation of the kinetics of virus capsid self-assembly. Biophys. J. 75, 2626–2636 (1998).PMC12999389826587Hagan M. F., Chandler D., Dynamic pathways for viral capsid assembly. Biophys. J. 91, 42–54 (2006).PMC147907816565055Jack R. L., Hagan M. F., Chandler D., Fluctuation–dissipation ratios in the dynamics of self-assembly. Phys. Rev. E 76, 021119 (2007).17930018Hagan M. F., Elrad O. M., Jack R. L., Mechanisms of kinetic trapping in self-assembly and phase transformation. J. Chem. Phys. 135, 104115 (2011).PMC329259321932884Rapaport D., The role of reversibility in viral capsid growth: A paradigm for self-assembly. Phys. Rev. Lett. 101, 186101 (2008).18999841Rapaport D. C., Studies of reversible capsid shell growth. J. Phys.: Condens. Matter 22, 104115 (2010).21389449Rapaport D. C., Molecular dynamics simulation of reversibly self-assembling shells in solution using trapezoidal particles. Phys. Rev. E 86, 051917 (2012).23214824Whitelam S., et al. , The impact of conformational fluctuations on self-assembly: Cooperative aggregation of archaeal chaperonin proteins. Nano Lett. 9, 292–297 (2009).PMC267044319072304Nguyen H. D., Reddy V. S., Brooks C. L., Deciphering the kinetic mechanism of spontaneous self-assembly of icosahedral capsids. Nano Lett. 7, 338–344 (2007).17297998Nguyen H., Brooks C., Generalized structural polymorphism in self-assembled viral particles. Nano Lett. 8, 4574 (2008).PMC277218219367856Nguyen H. D., Reddy V. S., Brooks C. L., Invariant polymorphism in virus capsid assembly. J. Am. Chem. Soc. 131, 2606–2614 (2009).PMC276826319199626Wilber A. W., et al. , Reversible self-assembly of patchy particles into monodisperse icosahedral clusters. J. Chem. Phys. 127, 085106 (2007).17764305Wilber A. W., Doye J. P. K., Louis A. A., Lewis A. C. F., Monodisperse self-assembly in a model with protein-like interactions. J. Chem. Phys. 131, 175102 (2009).19895043Cheng S., Aggarwal A., Stevens M. J., Self-assembly of artificial microtubules. Soft Matter 8, 5666 (2012).Hagan M. F., Modeling viral capsid assembly. Adv. Chem. Phys. 155, 1–68 (2014).PMC431812325663722Whitelam S., Jack R. L., The statistical mechanics of dynamic pathways to self-assembly. Ann. Rev. Phys. Chem. 66, 143–163 (2015).25493714Seung H. S., Nelson D. R., Defects in flexible membranes with crystalline order. Phys. Rev. A 38, 1005–1018 (1988).9900464Kléman M., Curved crystals, defects and disorder. Adv. Phys. 38, 605–667 (1989).Meek D., Walton D., On surface normal and Gaussian curvature approximations given data sampled from a smooth surface. Comput. Aided Geomet. Des. 17, 521–543 (2000).Grason G. M., Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations. Phys. Rev. E 85, 031603 (2012).22587104Schneider S., Gompper G., Shapes of crystalline domains on spherical fluid vesicles. Europhys. Lett. 70, 136 (2005).Meng G., Paulose J., Nelson D. R., Manoharan V. N., Elastic instability of a crystal growing on a curved surface. Science 343, 634–637 (2014).24503849Grason G. M., Perspective: Geometrically frustrated assemblies. J. Chem. Phys. 145, 110901 (2016).Yang Y., Meyer R., Hagan M. F., Self-limited self-assembly of chiral filaments. Phys. Rev. Lett. 104, 258102 (2010).20867417Saranathan V., Narayanan S., Sandy A., Dufresne E. R., Prum R. O., Evolution of single gyroid photonic crystals in bird feathers. Proc. Natl. Acad. Sci. U.S.A. 118, e2101357118 (2021).PMC820185034074782Z. A. Almsherqi, T. Landh, S. D. Kohlwein, Y. Deng, “Chapter 6: Cubic membranes: The missing dimension of cell membrane organization” in International Review of Cell and Molecular Biology, International Review of Cell and Molecular Biology (Academic Press, 2009), vol. 274, pp. 275–342.PMC710503019349040Selstam E., Schelin J., Williams W. P., Brain A. P., Structural organisation of prolamellar bodies (PLB) isolated from Zea mays. parallel TEM, SAXS and absorption spectra measurements on samples subjected to freeze–thaw, reduced pH and high-salt perturbation. Biochim. Biophys. Acta (BBA): Biomemb. 1768, 2235–2245 (2007).17559801D. M. Anderson, H. T. Davis, J. C. C. Nitsche, L. E. Scriven, “Periodic surfaces of prescribed mean curvature” in Physics of Amphiphilic Layers, J. Meunier, D. Langevin, N. Boccara, Eds. (Springer, Berlin/Heidelberg, Berlin/Heidelberg, Germany, 1987), p. 130.Grosse-Brauckmann K., Gyroids of constant mean curvature. Exp. Math. 6, 33–50 (1997).Q. M. Dowling et al., Hierarchical design of pseudosymmetric protein nanoparticles. bioRxiv [Preprint] (2023). 10.1101/2023.06.16.545393 (Accessed 18 June 2023).10.1101/2023.06.16.545393Zandi R., van der Schoot P., Reguera D., Kegel W., Reiss H., Classical nucleation theory of virus capsids. Biophys. J. 90, 1939–1948 (2006).PMC138677416387781Zandi R., Dragnea B., Travesset A., Podgornik R., On virus growth and form. Phys. Rep. 847, 1–102 (2020).Belcastro S.-M., Hull T. C., Modelling the folding of paper into three dimensions using affine transformations. Linear Algebra Appl. 348, 273–282 (2002).C. M. Duque et al., Limits of economy and fidelity for programmable assembly of size-controlled triply-periodic polyhedra. ScholarWorks. https://scholarworks.umass.edu/data/180/. Deposited 11 April 2024.PMC1106705938669182