Self-assembly of tetravalent Goldberg polyhedra from 144 small components

Journal name:
Nature
Volume:
540,
Pages:
563–566
Date published:
DOI:
doi:10.1038/nature20771
Received
Accepted
Published online

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.

At a glance

Figures

  1. MnL2n-type polyhedral metal–organic ligand complexes.
    Figure 1: MnL2n-type polyhedral metal–organic ligand complexes.

    a, Schematic representation of MnL2n complexes with the symmetry of Platonic or Archimedean solids. Each vertex represents a metal ion centre and each edge represents an organic ligand. Under the prerequisite that each vertex connects to four edges (because the metal ion used in this system (palladium(ii)) has a square planar coordination geometry), only five structures are allowed. MnL2n complexes with n = 6, 12, 24 and 30 have previously been synthesized. b, Molecular structures of the organic ligands. The bend angle of the ligand θ dictates the final self-assembled product. A very small change in θ can result in very different final products.

  2. X-ray crystallographic analysis of self-assembled product 3.
    Figure 2: X-ray crystallographic analysis of self-assembled product 3.

    a, MEM (maximum entropy method) electron density map (0.6 electrons per Å3) of M30L60 complex 3. Supplementary Video 1 shows a rotating electron density map. b, Enlarged view around a ligand in M30L60 with its MEM electron density map. The modelled ligand framework, metal ions and counter ions (BF4) agree well with the observed electron density. c, Entire view of the X-ray crystallographic structure of M30L60. Methoxy substituents and counter ions are omitted for clarity. Space group, Pbca; lattice parameters, a = 73.0 Å, b = 73.2 Å and c = 143.4 Å; data resolution, 1.95 Å; R(I > 2σ(I)), 0.2049; Rfree(I > 2σ(I)), 0.2391. d, Simplified image of the obtained structure. Each vertex represents a metal ion centre and each edge represents an organic ligand. The structure has polyhedral chirality; a pair of left-and right-hand forms is shown.

  3. Schematic representation of Goldberg polyhedra.
    Figure 3: Schematic representation of Goldberg polyhedra.

    a, Goldberg polyhedra consist of pentagons and hexagons. Each polyhedron is denoted by the locational relationship between the closest pair of pentagons. In this case, the polyhedron is denoted tri-G(2, 4) (h = 2, k = 4). b, Definition of T number. The T number represents the square of the distance between the two closest pentagons on the hexagonal network. From the Pythagorean theorem, T = |h + k|2 = [|h| + |k|cos(60°)]2 + [|k|sin(60°)]2 = h2 + hk + k2. c, Definition of Q number. By simple analogy with the T number, the Q number is defined by the square of the distance between the two closest triangles on the square grid network. df, Diagrams of tet-G(h, k) polyhedra. tet-G(1, 0) with Q = 1 is equivalent to an octahedron (d), tet-G(1, 1) with Q = 2 is a cuboctahedron (e) and tet-G(2, 0) with Q = 4 a rhombicuboctahedron (f). g, Diagram of tet-G(2, 1) and tet-G(1, 2) (Q = 5), which have topology that is identical to that of the left- and right-hand forms of M30L60, respectively, shown in Fig. 1a, b. The ‘asymmetric’ h and k values support the origin of chirality. h, Summary of extended Goldberg polyhedra ordered by Q number. Only those corresponding to Q = 1, 2 and 4 have been observed experimentally previously; in this work we observe those corresponding to Q = 5 and 8; and the others are predicted by theory, but have yet to be observed.

  4. X-ray crystallographic analysis of self-assembled product 4.
    Figure 4: X-ray crystallographic analysis of self-assembled product 4.

    a, Entire view of the MEM electron density map (0.6 electrons per Å3) for M48L96 complex 4 (right). This complex has the topology of the tet-G(2, 2) polyhedron with Q = 8 (left). Supplementary Video 2 shows a rotating electron density map. b, Enlarged view around a ligand in M48L96 with its MEM electron density map. The modelled ligand structure agrees well with the observed electron density. Space group, ; lattice parameters, a = b = 63.7 Å, c = 94.6 Å; data resolution, 2.85 Å; R(I > 2σ(I)), 0.2181; Rfree(I > 2σ(I)), 0.2465. c, Sliced image of the M48L96 complex with the MEM electron density map. No distinguishable peaks in electron density are observed in the void space; this supports the validity of the model building. d, Crystal structure of 4 emphasizing the tet-G(2, 2) topology and metal centres.

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

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

  7. 1H NMR of ligand 1.
    Extended Data Fig. 3: 1H NMR of ligand 1.

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

  8. 13C NMR of ligand 1.
    Extended Data Fig. 4: 13C NMR of ligand 1.

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

  9. 1H NMR of 3.
    Extended Data Fig. 5: 1H NMR of 3.

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

  10. 1H NMR of 3.
    Extended Data Fig. 6: 1H NMR of 3.

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

  11. 1H DOSY NMR of 3.
    Extended Data Fig. 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.

  12. 1H DOSY NMR of 3.
    Extended Data Fig. 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.

Tables

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

Videos

  1. MEM electron density map of M30L6.
    Video 1: MEM electron density map of M30L6.
    360° rotating movie of the electron density map.
  2. MEM electron density map of M48L96.
    Video 2: MEM electron density map of M48L96.
    360° rotating movie of the electron density map.

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Author information

Affiliations

  1. Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

    • Daishi Fujita,
    • Yoshihiro Ueda &
    • Makoto Fujita
  2. PRESTO (Precursory Research for Embryonic Science and Technology), Japan Science and Technology Agency, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

    • Daishi Fujita
  3. ACCEL, Japan Science and Technology Agency, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

    • Daishi Fujita,
    • Yoshihiro Ueda &
    • Makoto Fujita
  4. Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

    • Sota Sato
  5. ERATO, Japan Science and Technology Agency, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

    • Sota Sato
  6. Japan Synchrotron Radiation Research Institute (JASRI), SPring-8, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan

    • Nobuhiro Mizuno &
    • Takashi Kumasaka

Contributions

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.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

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

Author details

Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: NMR study of the self-assembly of ligand 1. (69 KB)

    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.

  2. Extended Data Figure 2: Schematic representation of larger tet-G(h, k) polyhedra. (214 KB)

    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.

  3. Extended Data Figure 3: 1H NMR of ligand 1. (72 KB)

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

  4. Extended Data Figure 4: 13C NMR of ligand 1. (67 KB)

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

  5. Extended Data Figure 5: 1H NMR of 3. (69 KB)

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

  6. Extended Data Figure 6: 1H NMR of 3. (96 KB)

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

  7. Extended Data Figure 7: 1H DOSY NMR of 3. (73 KB)

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

  8. Extended Data Figure 8: 1H DOSY NMR of 3. (81 KB)

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

Extended Data Tables

  1. Extended Data Table 1: Crystal data and structural refinement for M30L60 (3) and M48L96 (4) (265 KB)

Supplementary information

Video

  1. Video 1: MEM electron density map of M30L6. (50.98 MB, Download)
    360° rotating movie of the electron density map.
  2. Video 2: MEM electron density map of M48L96. (47.21 MB, Download)
    360° rotating movie of the electron density map.

Crystallographic information files

  1. Supplementary Data 1 (13.4 MB)

    X-ray data of M30L60.

  2. Supplementary Data 2 (1.1 MB)

    X-ray data of M48L96.

Additional data