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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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).
The authors declare no competing financial interests.
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
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.
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.
Full range spectrum: 500 MHz, CDCl3, 27 °C.
Full range spectrum: 125 MHz, CDCl3, 27 °C.
Full range spectrum: 500 MHz, DMSO-d6, 27 °C.
Full range spectrum: 500 MHz, DMF-d5, 27 °C.
Full range spectrum: 500 MHz, DMSO-d6, 27 °C. For comparison with M12L24 or M24L48 complexes, see ref. 36.
Full range spectrum: 500 MHz, DMF-d5, 27 °C. For comparison with M12L24 or M24L48 complexes, see ref. 36.
X-ray data of M30L60. (CIF 13745 kb)
X-ray data of M48L96. (CIF 1216 kb)
360° rotating movie of the electron density map. (MOV 52204 kb)
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). https://doi.org/10.1038/nature20771
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