A guide to bullvalene stereodynamics

Abstract

Here, we analyze the stereodynamic properties of bullvalenes using principal moments of inertia and exit vector plots to draw comparisons with commonly used ring systems in medicinal chemistry. To aid analyses, we first classify (i) the four elementary rearrangement steps available to substituted bullvalenes, which (ii) can be described by applying positional descriptors (α, β, γ, and δ) to the substituents. We also (iii) derive an intuitive equation to calculate the number of isomers for a given bullvalene system. Using DFT-modelled structures for di-, tri-, and tetrasubstituted bullvalenes, generated using a newly developed computational tool (bullviso), we show that their 3D shapes and the exit vectors available from the bullvalene scaffold make them comparable to other bioisosteres currently used to replace planar aromatic ring systems in drug discovery. Unlike conventional ring systems, the shapeshifting valence isomerism of bullvalenes gives rise to numerous shapes and substituent relationships attainable as a concentration-independent dynamic covalent library from a single compound. We visualize this property by applying population weightings to the principal moments of inertia and exit vector analyses to reflect the relative thermodynamic stabilities of the available isomers.

Fig. 1. (a) The BV isomer barcode labelling system, top, and relative positional labels, bottom. (b) The possible exchange processes following one Cope rearrangement step, enumerated for each BV position. Full isomerization requires sequential steps that include (P = participating) and exclude (NP = non-participating) the substituent in the rearranging 1,5-hexadiene motif (shown in orange). (c) Illustration of the higher symmetry in the transition state for γ_P ⇌ γ_P Cope rearrangement relative to the ground state. (d) The partial isomer network of a heterodisubstituted BV showing the positional exchange arising from three sequential Cope rearrangement steps.

Fig. 2. Structural formulae and isomer barcodes of (a) one of the 240 unique isomers of BV 1, showing the calculation of N_iso, and (b) the three C₃-symmetric isomers of BV 2 that are accounted for by a correction factor of S = 3.

Fig. 3. (a) Structural formulae of methyl-substituted BVs. (b) The population-weighted isomer interconversion network calculated for Me₂BV (PBE0-D3/def2-SV(P)). The diagram has a mirror plane with achiral isomers down the middle and enantiomeric pairs of chiral structures on either side. Chiral structures are labelled with an R/S descriptor according to the stereogenic α position. (c) A graph of the relative energies of Me₂BV isomers. Pairs of enantiomers are isoenergetic, so are represented just once.

Fig. 4. (a–c) PMI plots for the shapeshifting networks of (a) Me₂BV, (b) Me₃BV, and (c) Me₄BV. Substituent positional labels are given for Me₂BV. For clarity, these labels are not shown on the plots for Me₃BV and Me₄BV. See Tables S6 and S7 for labelled data. (d–e) Population-weighted PMI plots for the shapeshifting networks of (d) Me₂BV, (e) Me₃BV, and (f) Me₄BV where the data points are scaled by calculated Boltzmann distributions at 298 K (PBE0-D3/def2-SV(P)). The modelled structure of the lowest-energy isomer for each BV is shown inset. (g) An overlay of the PMI plots of Me₂BV (blue), Me₃BV (orange), and Me₄BV (purple) showing that none of the isomers have ΣNPR values close to the rod–disc axis. (h) A PMI plot for common ring systems. See Table S8 for compound identities. (i) Structural formulae and ΣNPR values for 1,4-dimethyladamantane (3), 1,2-dimethylcubane (4), and 1,4-dimethylcubane (5).

Fig. 5. (a) The vectors v₁ and v₂ for two substituent attachment points on a BV (shown for (αS)-γ,δ′-Me₂BV) which are defined by (b) the geometric parameters r, φ₁, φ₂, and θ.

Fig. 6. (a–c) Distance–dihedral angle EV plots and (d–i) Boltzmann population-weighted distance–dihedral angle EV plots (298 K, PBE0-D3/def2-SV(P)) for the isomers of (a, d and g) Me₂BV, (b, e and h) Me₃BV, and (c, f and i) Me₄BV. (j) Overlaid distance versus dihedral angle EV plot of all three methyl-substituted BVs. (g–j) Plots include characteristic areas of EV plots in grey that correspond to those found in disubstituted cycloalkanes, a = cis-1,2-disubstituted cyclopropanes, b = cis-1,3-disubstituted aliphatic rings and cis-1,4-disubstituted cyclohexanes, c = trans-1,4-disubstituted cyclohexanes, d = trans-1,3-disubstituted cyclopentanes and cyclohexanes.

Fig. 7. The plane angles subtended by C–Me EVs in the nine lowest energy isomers of Me₂BV. Data points are scaled according to the Boltzmann population at 298 K (PBE0-D3/def2-SV(P)).