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What tangled webs we weave

Nature Chemistryvolume 10pages10781079 (2018) | Download Citation

Knots have been rigorously studied since the 1860s, but only in the past 30 years have they been made in the laboratory in molecular form. Now, the most complex small-molecule examples so far — a composite knot and an isomeric link, each with nine crossings — have been prepared.

The 2016 Nobel Prizes for both chemistry and physics were awarded for work that involved topological aspects of materials — these are heady days for applications of topology. Significant advances in the synthesis of molecules with non-trivial topologies have been realized in recent years. Just a decade ago, however, it was an open problem in the field of chemical topology whether chemists could prepare small-molecule prime knots more complex than the unknot and trefoil (Fig. 1a,b)1. Since that time, several complex knots have been synthesized, including the pentafoil and 819 knots (Fig. 1c,d) through rational design by Leigh and co-workers2 and, more recently, serendipitous complex knot formation by Kim and colleagues3. Now, writing in Nature Chemistry, a team led by David Leigh reports4 the synthesis of the most complex composite knot and [3]catenane ever prepared — each of which contains nine crossings. The molecular composite knot (Fig. 1f) is impressively large, with a backbone of 324 atoms, and each of the twisted macrocycles in the [3]catenane (Fig. 1g) is a 108-membered ring.

Fig. 1: Molecular knots and links that have been synthesized.
Fig. 1

a, Unknot (trivial knot) 01. b, Trefoil knot 31. c, Pentafoil knot 51. d, Non-alternating 819 knot. e, Square knot +31#–31. f, Leigh’s composite knot +31#+31#+31. g, Link \({9}_7^3\), a twisted cyclic [3]catenane. Panels f and g adapted from ref. 4, Springer Nature Ltd.

Knot theory is a specialized subfield of topology. The origins of modern knot theory can be traced to the nineteenth-century work of four physicists: Herman Von Helmholtz, William Thomson (Lord Kelvin), James Clerk Maxwell and Peter Guthrie Tait5. Interest in studying knots intensified in the 1860s with the development of Thompson’s vortex theory, which posited that each chemical element was a different knot in the ether. This encouraged Tait and others to begin a systematic enumeration of knots5. It is also appropriate at this point to recall Tait’s warning6 that: “…the subject is a very much more difficult and intricate one than at first sight one is inclined to think.” Some of these intricacies will be unpacked here, but for a complete treatment of knot theory and how it relates to molecular knots, the interested reader is directed to a recent review article2.

Knots are mathematically defined as a closed loop of a single component that is embedded in three-dimensional space. Knots are characterized in many ways: including their number of crossings; whether they are prime or composite; chiral or amphicheiral (achiral); or have alternating (Fig. 1b,c) or non-alternating crossings (Fig. 1d,e). A composite knot can be decomposed into two or more non-trivial factor knots, whereas a prime knot cannot. This is analogous to a composite number, which has other factors besides 1 and itself, whereas a prime number does not. For example, the square knot has six crossings and is formed by combining two enantiomeric trefoil knots (Fig. 1e). The nomenclature of composite knots uses the hash symbol between the two factor knots; for example, the square knot is represented by +31#–31 where the plus or minus symbol indicates the chirality of each sub-component knot.

A new era in the field of molecular topology began in 1989 when a team led by Sauvage reported the synthesis of the first small-molecule trefoil knot7. This group also reported8 the first synthesis of a composite knot, which was formed in 2.5% yield as part of a mixture that contained all possible stereoisomers, namely the meso square knot (+31#–31) as well as two enantiomeric granny knots (+31#+31 and –31#–31). This result was at the leading edge of the field in 1996, but tremendous advances have been made since then. The new composite knot now reported4 by Leigh and co-workers was synthesized in a stereoselective fashion, giving only the +31#+31#+31 (from the homochiral Δ stereochemistry of the six iron centres) and –31#–31#–31 (from the homochiral Λ stereochemistry of the six iron centres) isomers in high yields. The other possible stereoisomers (+31#+31#–31 or –31#–31#+31) were not observed.

These composite knots are akin to the ones we tie with shoelaces or ropes. A square knot is formed by crossing the right strand over the left followed by left over right. A granny knot is formed by crossing right over left (or left over right) two times. Sometimes novice knot tiers want to make a knot more secure, and so they cross right over left three times — this is Leigh’s composite knot if the loose ends were joined.

Leigh and colleagues used the circular helicate method to make these interwoven molecules — whereby organic ligands are gathered together and assembled into a superstructure by coordination with metal ions — which has been highly successful for the synthesis of other torus topologies such as the pentafoil and 819 knot (Fig. 1) as well as the Star of David link2. The organic ligand was carefully designed to have an angled extension and limited conformational flexibility and was assembled into a hexameric circular helicate comprising six ligands and six iron(ii) ions in 90% yield. Ring-closing metathesis on a dilute solution — the preferred method for connecting the loose ends in many interlocked system s — worked remarkably well. The 12 loose ends were linked in one fashion to give the metal complex of the composite knot and in another to give the metal complex of the isomeric \({9}_7^3\) link, which can also be described as a twisted-ring cyclic [3]catenane, in 92% combined yield. It is common for isomers to be formed in the course of preparing molecules that have complex topologies, and similar amounts of each were produced. From a topological perspective, these species are not technically a knot or link because of the presence of the metal ions. Nevertheless, demetallation to form the tri-trefoil knot and link (Figs. 1f,g) was accomplished by treatment with sodium hydroxide.

The separation of the knot and link mixture proved difficult, and size-exclusion, gel permeation, and standard and chiral high-performance liquid chromatography all failed. Crystallization, perhaps the oldest separation method, did work — the knot–metal complex crystallized from the mixture as the hexafluorophosphate salt. This was especially beneficial because the characterization data (NMR spectroscopy and mass spectrometry) for the knot–link mixture could then be compared to the data for the pure knot. Tandem mass spectrometry (MS-MS) experiments were particularly useful and the resulting fragmentation patterns enabled each product to be identified. Of note is the unique fragmentation of catenanes, which lose a macrocycle upon bond cleavage and unthreading. These analytical experiments established the topology of the products, but additional proof was provided by X-ray diffraction of the crystals of the knot–metal complex. To solve the structure, a rigid-body approach that is similar to methods used for protein crystallography had to be employed but, importantly, the refined structure confirmed the topology of the composite knot.

The future for chemical topology is bright and it will be fascinating to see what comes next. But why synthesize molecular knots? Part of the draw is the intrinsic beauty of the structures, but also the significant synthetic challenge they pose. The circular helicate method provides elegant routes to torus knots and links2, but it is unclear if a general synthetic method can be devised for non-torus systems. In some ways, molecular knots and links provide a new frontier for synthetic chemists to conquer — much as complex natural products did 50 years ago. In both cases, stereochemical control is of paramount importance; however, with natural products it is Euclidean stereochemistry (stereogenic centres, planes and axes), whereas with knots and links there is topological stereochemistry to consider as well9,10.


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  1. Department of Chemistry at Franklin & Marshall College, Lancaster, PA, USA

    • Edward E. Fenlon


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