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Self-assembly of tetravalent Goldberg polyhedra from 144 small components


Rational control of the self-assembly of large structures is one of the key challenges in chemistry1,2,3,4,5,6,7,8,9, and is believed to become increasingly difficult and ultimately impossible as the number of components involved increases. So far, it has not been possible to design a self-assembled discrete molecule made up of more than 100 components. Such molecules—for example, spherical virus capsids10—are prevalent in nature, which suggests that the difficulty in designing these very large self-assembled molecules is due to a lack of understanding of the underlying design principles. For example, the targeted assembly of a series of large spherical structures containing up to 30 palladium ions coordinated by up to 60 bent organic ligands11,12,13,14,15,16 was achieved by considering their topologies17. Here we report the self-assembly of a spherical structure that also contains 30 palladium ions and 60 bent ligands, but belongs to a shape family that has not previously been observed experimentally17. The new structure consists of a combination of 8 triangles and 24 squares, and has the symmetry of a tetravalent Goldberg polyhedron18,19. Platonic and Archimedean solids have previously been prepared through self-assembly, as have trivalent Goldberg polyhedra, which occur naturally in the form of virus capsids20 and fullerenes21. But tetravalent Goldberg polyhedra have not previously been reported at the molecular level, although their topologies have been predicted using graph theory. We use graph theory to predict the self-assembly of even larger tetravalent Goldberg polyhedra, which should be more stable, enabling another member of this polyhedron family to be assembled from 144 components: 48 palladium ions and 96 bent ligands.

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Figure 1: MnL2n-type polyhedral metal–organic ligand complexes.
Figure 2: X-ray crystallographic analysis of self-assembled product 3.
Figure 3: Schematic representation of Goldberg polyhedra.
Figure 4: X-ray crystallographic analysis of self-assembled product 4.


  1. 1

    Conn, M. M. & Rebek, J. Self-assembling capsules. Chem. Rev. 97, 1647–1668 (1997)

    CAS  Article  Google Scholar 

  2. 2

    Caulder, D. L. & Raymond, K. N. Supermolecules by design. Acc. Chem. Res. 32, 975–982 (1999)

    CAS  Article  Google Scholar 

  3. 3

    Saalfrank, R. W., Maid, H. & Scheurer, A. Supramolecular coordination chemistry: the synergistic effect of serendipity and rational design. Angew. Chem. Int. Ed. 47, 8794–8824 (2008)

    CAS  Article  Google Scholar 

  4. 4

    Forgan, R. S., Sauvage, J.-P. & Stoddart, J. F. Chemical topology: complex molecular knots, links, and entanglements. Chem. Rev. 111, 5434–5464 (2011)

    CAS  Article  Google Scholar 

  5. 5

    Chakrabarty, R., Mukherjee, P. S. & Stang, P. J. Supramolecular coordination: self-assembly of finite two- and three-dimensional ensembles. Chem. Rev. 111, 6810–6918 (2011)

    CAS  Article  Google Scholar 

  6. 6

    Young, N. J. & Hay, B. P. Structural design principles for self-assembled coordination polygons and polyhedra. Chem. Commun. 49, 1354–1379 (2013)

    CAS  Article  Google Scholar 

  7. 7

    Smulders, M. M. J., Riddell, I. A., Browne, C. & Nitschke, J. R. Building on architectural principles for three-dimensional metallosupramolecular construction. Chem. Soc. Rev. 42, 1728–1754 (2013)

    CAS  Article  Google Scholar 

  8. 8

    Cook, T. R. & Stang, P. J. Recent developments in the preparation and chemistry of metallacycles and metallacages via coordination. Chem. Rev. 115, 7001–7045 (2015)

    CAS  Article  Google Scholar 

  9. 9

    Hsia, Y. et al. Design of a hyperstable 60-subunit protein icosahedron. Nature 535, 136–139 (2016)

    CAS  ADS  Article  Google Scholar 

  10. 10

    Zlotnick, A. To build a virus capsid: an equilibrium model of the self assembly of polyhedral protein complexes. J. Mol. Biol. 241, 59–67 (1994)

    CAS  Article  Google Scholar 

  11. 11

    Harris, K., Fujita, D. & Fujita, M. Giant hollow MnL2n spherical complexes: structure, functionalisation and applications. Chem. Commun. 49, 6703–6712 (2013)

    CAS  Article  Google Scholar 

  12. 12

    Fujita, D., Yokoyama, H., Ueda, Y., Sato, S. & Fujita, M. Geometrically restricted intermediates in the self-assembly of an M12L24 cuboctahedral complex. Angew. Chem. Int. Ed. 54, 155–158 (2015)

    CAS  Article  Google Scholar 

  13. 13

    Suzuki, K., Tominaga, M., Kawano, M. & Fujita, M. Self-assembly of an M6L12 coordination cube. Chem. Commun. 1638–1640 (2009)

  14. 14

    Tominaga, M. et al. Finite, spherical coordination networks that self-organize from 36 small components. Angew. Chem. Int. Ed. 43, 5621–5625 (2004)

    CAS  Article  Google Scholar 

  15. 15

    Sun, Q. F. et al. Self-assembled M24L48 polyhedra and their sharp structural switch upon subtle ligand variation. Science 328, 1144–1147 (2010)

    CAS  ADS  Article  Google Scholar 

  16. 16

    Fujita, D. et al. Self-assembly of M30L60 icosidodecahedron. Chem 1, 91–101 (2016)

    CAS  Article  Google Scholar 

  17. 17

    Coxeter, H. S. M. Introduction to Geometry 2nd edn, Ch. 10, 21 (Wiley, 1989)

  18. 18

    Goldberg, M. A class of multi-symmetric polyhedra. Tohoku Math. J. 43, 104–108 (1937)

    MATH  Google Scholar 

  19. 19

    Schein, S. & Gayed, J. M. Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses. Proc. Natl Acad. Sci. USA 111, 2920–2925 (2014)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  20. 20

    Caspar, D. L. D. & Klug, A. Physical principles in the construction of regular viruses. Cold Spring Harb. Symp. Quant. Biol. 27, 1–24 (1962)

    CAS  Article  Google Scholar 

  21. 21

    Fowler, P. W. & Manolopoulos, D. E. An Atlas of Fullerenes Ch. 2 (Courier Corporation, 2006)

  22. 22

    Bunzen, M. et al. Self-assembly of M24L48 polyhedra based on empirical prediction. Angew. Chem. Int. Ed. 51, 3161–3163 (2012)

    CAS  Article  Google Scholar 

  23. 23

    Takata, M. The MEM/Rietveld method with nano-applications — accurate charge-density studies of nano-structured materials by synchrotron-radiation powder diffraction. Acta Crystallogr. A 64, 232–245 (2008)

    CAS  ADS  Article  Google Scholar 

  24. 24

    Fujita, D. et al. Protein encapsulation within synthetic molecular hosts. Nat. Commun. 3, 1093 (2012)

    ADS  Article  Google Scholar 

  25. 25

    Prasad, B. V. & Schmid, M. F. Principles of virus structural organization. Adv. Exp. Med. Biol. 726, 17–47 (2012)

    CAS  Article  Google Scholar 

  26. 26

    Sikirić, M. D. & Deza, M. in The Mathematics and Topology of Fullerenes (eds Cataldo, F. et al.) Ch. 6, 103–116 (Springer, 2011)

    Article  Google Scholar 

  27. 27

    Dutour, M. & Deza, M. Goldberg–Coxeter construction for 3- and 4-valent plane graphs. Electron. J. Combin. 11, R20 (2004)

    MATH  Google Scholar 

  28. 28

    Brinkmann, G. & Deza, M. Lists of face-regular polyhedra. J. Chem. Inf. Comput. Sci. 40, 530–541 (2000)

    CAS  Article  Google Scholar 

  29. 29

    Zhou, Z. & Harris, K. D. M. Design of a molecular quasicrystal. ChemPhysChem 7, 1649–1653 (2006)

    CAS  Article  Google Scholar 

  30. 30

    Vekilov, Y. K. & Chernikov, M. A. Quasicrystals. Phys. Uspekhi 53, 537–560 (2010)

    CAS  ADS  Article  Google Scholar 

  31. 31

    Kabsch, W. XDS. Acta Crystallogr. D 66, 125–132 (2010)

    CAS  Article  Google Scholar 

  32. 32

    Sheldrick, G. M. SHELXT — integrated space-group and crystal-structure determination. Acta Crystallogr. A 71, 3–8 (2015)

    Article  Google Scholar 

  33. 33

    Sheldrick, G. M. Crystal structure refinement with SHELXL. Acta Crystallogr. C 71, 3–8 (2015)

    Article  Google Scholar 

  34. 34

    Tanaka, H. et al. ENIGMA: maximum-entropy method program package for huge systems. J. Appl. Cryst. 35, 282–286 (2002)

    CAS  Article  Google Scholar 

  35. 35

    Schwerdtfeger, P., Wirz, L. N. & Avery, J. The topology of fullerenes. WIREs Comp. Mol. Sci. 5, 96–145 (2015)

    CAS  Article  Google Scholar 

  36. 36

    Yokoyama, H., Ueda, Y., Fujita, D., Sato, S. & Fujita, M. Finely resolved threshold for the Sharp M12L24/M24L48 structural switch in multi-component MnL2n polyhedral assemblies: X-ray, MS, NMR, and ultracentrifugation analyses. Chem. Asian J. 10, 2292–2295 (2015)

    CAS  Article  Google Scholar 

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This work was supported by the JST–PRESTO programme and the ACCEL programme, and partly supported by KAKENHI (25102007). The synchrotron X-ray crystallography was performed at the BL38B1 and BL41XU beamlines at SPring-8 (2014A0042 and 2015B0120); preliminarily experiments were performed at the BL-1A beamline at KEK PF (2015G097). We thank M. Kotani and M. Tagami for providing us with mathematical discussion (see Methods) and references (refs 26, 27, 28).

Author information




D.F. and Y.U. performed and analysed the experiments. D.F. and S.S. worked on the preliminary X-ray diffraction analysis. N.M. and T.K. finalized the X-ray data. D.F. built the mathematical discussion. D.F. and M.F. wrote the paper. D.F. and M.F. designed and supervised the research project.

Corresponding authors

Correspondence to Daishi Fujita or Makoto Fujita.

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Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks F. Beuerle, P. Schwerdtfeger and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 NMR study of the self-assembly of ligand 1.

a, 1H NMR spectrum (500 MHz, DMSO-d6, 300 K) of ligand 1. The signal denoted PyHα is derived from the protons in the pydridyl α-position; that denoted PyHβ is derived from the protons in the pyridyl β-position. The signal denoted –OCH3 is from the methoxy protons. b, 1H NMR spectrum of 1 after self-assembly with palladium(ii) ions (BF4 salt). The aromatic signals are shifted downfield and heavily broadened. c, 1H DOSY spectrum of 1 after self-assembly with palladium(ii) ions (BF4 salt). The spectrum indicates a single product with a diffusion coefficient D of 2.6 × 10−11 m2 s−1 (logD = −10.58). The grey band is a guide to the eye. All of the NMR spectra (500 MHz) were measured for DMSO-d6 solutions at 300 K.

Extended Data Figure 2 Schematic representation of larger tet-G(h, k) polyhedra.

Polyhedra with the topology of tet-G(3, 0) (or, equivalently of tet-G(0, 3); Q = 9), tet-G(1, 3) (Q = 10) or tet-G(2, 3) (Q = 13). For Q = 9 and Q = 10, the other structure in the chiral pair (tet-G(3, 1) and tet-G(3, 2), respectively) is a mirror image of the polyhedron shown.

Extended Data Figure 3 1H NMR of ligand 1.

Full range spectrum: 500 MHz, CDCl3, 27 °C.

Extended Data Figure 4 13C NMR of ligand 1.

Full range spectrum: 125 MHz, CDCl3, 27 °C.

Extended Data Figure 5 1H NMR of 3.

Full range spectrum: 500 MHz, DMSO-d6, 27 °C.

Extended Data Figure 6 1H NMR of 3.

Full range spectrum: 500 MHz, DMF-d5, 27 °C.

Extended Data Figure 7 1H DOSY NMR of 3.

Full range spectrum: 500 MHz, DMSO-d6, 27 °C. For comparison with M12L24 or M24L48 complexes, see ref. 36.

Extended Data Figure 8 1H DOSY NMR of 3.

Full range spectrum: 500 MHz, DMF-d5, 27 °C. For comparison with M12L24 or M24L48 complexes, see ref. 36.

Extended Data Table 1 Crystal data and structural refinement for M30L60 (3) and M48L96 (4)

Supplementary information

Supplementary Data 1

X-ray data of M30L60. (CIF 13745 kb)

Supplementary Data 2

X-ray data of M48L96. (CIF 1216 kb)

MEM electron density map of M30L6.

360° rotating movie of the electron density map. (MOV 52204 kb)

MEM electron density map of M48L96.

360° rotating movie of the electron density map. (MOV 48346 kb)

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Fujita, D., Ueda, Y., Sato, S. et al. Self-assembly of tetravalent Goldberg polyhedra from 144 small components. Nature 540, 563–566 (2016).

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