This is one of those posts of a molecule whose very structure is interesting enough to merit a picture and a 3D model. The study[1] reports a molecular knot with the remarkable number of eight crossings.
The DOI for the 3D model is 10.5517/CCDC.CSD.CC1M85Y0Â (or click on the image above). Such topology intersects with work we did a few years back on high-orderÂ crossings in fully conjugatedÂ Ï€-systems[2], which were then illustrated[3] with hypothetical charged higher order annulenes exhibiting linking numbers LkÂ of up to 6Ï€. A fullyÂ Ï€-conjugated system, also with a linking number in the Ï€-frameworkÂ ofÂ 6Ï€ but in the form of a trefoil braid was even suggested on this blog, with a common feature of a central templating atom (a cation rather than an anion). Another example of a previously reportedÂ pentadecanuclear manganese metallacycle[4] was also assigned a linking number ofÂ 6Ï€.
The molecule above is not completelyÂ Ï€-conjugated around the braid and so special properties related to aromaticity and associated ring currents resulting from the topology of the cyclic conjugation[5] are not expected to accrue in the eight-crossing molecular braid[1]. We might also look forward to examples of the characterisation ofÂ braids with an odd-number of crossings such as trefoils, pentafoils, heptafoils, etc, as associated with the nameÂ MÃ¶bius.
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References
- J.J. Danon, A. KrÃ¼ger, D.A. Leigh, J. Lemonnier, A.J. Stephens, I.J. Vitorica-Yrezabal, and S.L. Woltering, "Braiding a molecular knot with eight crossings", Science, vol. 355, pp. 159-162, 2017. http://dx.doi.org/10.1126/science.aal1619
- S.M. Rappaport, and H.S. Rzepa, "Intrinsically Chiral Aromaticity. Rules Incorporating Linking Number, Twist, and Writhe for Higher-Twist MoÌˆbius Annulenes", Journal of the American Chemical Society, vol. 130, pp. 7613-7619, 2008. http://dx.doi.org/10.1021/ja710438j
- C.S. Wannere, H.S. Rzepa, B.C. Rinderspacher, A. Paul, C.S.M. Allan, H.F. Schaefer, and P.V.R. Schleyer, "The Geometry and Electronic Topology of Higher-Order Charged MoÌˆbius Annulenesâ€ ", The Journal of Physical Chemistry A, vol. 113, pp. 11619-11629, 2009. http://dx.doi.org/10.1021/jp902176a
- H.S. Rzepa, "Linking Number Analysis of a Pentadecanuclear Metallamacrocycle: A MoÌˆbius-Craig System Revealed", Inorganic Chemistry, vol. 47, pp. 8932-8934, 2008. http://dx.doi.org/10.1021/ic800987f
- P.L. Ayers, R.J. Boyd, P. Bultinck, M. Caffarel, R. CarbÃ³-Dorca, M. CausÃ¡, J. Cioslowski, J. Contreras-Garcia, D.L. Cooper, P. Coppens, C. Gatti, S. Grabowsky, P. Lazzeretti, P. Macchi, Ã. MartÃn PendÃ¡s, P.L. Popelier, K. Ruedenberg, H. Rzepa, A. Savin, A. Sax, W.E. Schwarz, S. Shahbazian, B. Silvi, M. SolÃ , and V. Tsirelson, "Six questions on topology in theoretical chemistry", Computational and Theoretical Chemistry, vol. 1053, pp. 2-16, 2015. http://dx.doi.org/10.1016/j.comptc.2014.09.028
A hexafoil knot (Star of David) is reported by the Leigh group, DOI: 10.1038/nchem.2056, 3D Model DOI: 10.5517/CC129N45.
A pentafoil knot is also reported, DOI: 10.1126/science.aaf3673, 3D Model DOI: 10.5517/CCDC.CSD.CC1KFN3R.
In another post, I noted Derek Lowe’s book on 250 milestones in chemistry, highlighting two entries. As the same time, I also got Clifford Pickover’s book on 250 milestones in mathematics. You might expect that knots feature in this book. Again, I note two interesting entries.
1. Pickover places the discovery of knots at around 100,000 BC. He also mentions (and he was writing in 2009) that around 1.7 million non-equivalent knots with 16 or fewer crossings have been discovered.
2. His second milestone dates from 1988 when Sumners and Whittington (the latter a chemist) modelled ropes and other objects such as polymer chains, finding using purely mathematical procedures that nearly all sufficiently long self avoiding random walks will contain a knot (DOI: 10.1088/0305-4470/21/7/030), or more specifically that for n steps in a random walk, the knot probability goes to unity as n goes to ∞.
3. Finally, I note that Wikipedia has a whole section on knotted proteins, including trefoils. The topic of knotted proteins is a recent one, the first suggestions being in 1994!
A search of the protein databank (PDB) as http://www.rcsb.org/pdb/search/advSearch.do?search=new reveals 39 systems described by the term trefoil!