Molecular knot

Organic molecules containing knots may fall into the categories of slipknots or pseudo-knots.[2] They are not considered mathematical knots because they are not a closed curve, but rather a knot that exists within an otherwise linear chain, with termini at each end. Knotted proteins are thought to form molecular knots during their tertiary structure folding process, and knotted nucleic acids generally form molecular knots during genomic replication and transcription,[6] though details of knotting mechanism continue to be disputed and ambiguous. Molecular simulations are fundamental to the research on molecular knotting mechanisms.

Knotted DNA was found first by Liu et al. in 1981, in single-stranded, circular, bacterial DNA, though double-stranded circular DNA has been found to also form knots. Naturally knotted RNA has not yet been reported.[7]

A number of proteins containing naturally occurring molecular knots have been identified. The knot types found to be naturally occurring in proteins are the ${\displaystyle +3_{1},-3_{1},4_{1},-5_{2},}$ and ${\displaystyle +6_{1}}$knots, as identified in the KnotProt database of known knotted proteins.[8]

Several synthetic molecular knots have been reported.[9][10][11][12][13][14][15]

Crystal structure of a contra-helical trefoil knot reported by Zhichang Liu and coworkers in Nat. Synth. 2023, 2, 17–25[15]

Knot types that have been successfully synthesized in molecules are ${\displaystyle 3_{1},4_{1},5_{1}}$and 8₁₉ knots. Though the ${\displaystyle -5_{2}}$ and ${\displaystyle +6_{1}}$ knots have been found to naturally occur in knotted molecules, they have not been successfully synthesized. Small-molecule composite knots have also not yet been synthesized.[7]

Artificial DNA, RNA, and protein knots have been successfully synthesized. DNA is a particularly useful model of synthetic knot synthesis, as the structure naturally forms interlocked structures and can be easily manipulated into forming knots[16] control precisely the raveling necessary to form knots. Molecular knots are often synthesized with the help of crucial metal ion ligands.[7]

The first researcher to suggest the existence of a molecular knot in a protein was Jane Richardson in 1977, who reported that carbonic anhydrase B (CAB) exhibited apparent knotting during her survey of various proteins' topological behavior.[17] However, the researcher generally attributed with the discovery of the first knotted protein is Marc. L. Mansfield in 1994, as he was the first to specifically investigate the occurrence of knots in proteins and confirm the existence of the trefoil knot in CAB. Knotted DNA was found first by Liu et al. in 1981, in single-stranded, circular, bacterial DNA, though double-stranded circular DNA has been found to also form knots.[18]

In 1989, Sauvage and coworkers reported the first synthetic knotted molecule: a trefoil synthesized via a double-helix complex with the aid of Cu+ ions.[19]

Vogtle et al. was the first to describe molecular knots as knotanes in 2000.[1] Also in 2000 was William Taylor's creation of an alternative computational method to analyze protein knotting that set the termini at a fixed point far enough away from the knotted component of the molecule that the knot type could be well-defined. In this study, Taylor discovered a deep ${\displaystyle 4_{1}}$ knot in a protein.[20] With this study, Taylor confirmed the existence of deeply knotted proteins.

In 2007, Eric Yeates reported the identification of a molecular slipknot, which is when the molecule contains knotted subchains even though their backbone chain as a whole is unknotted and does not contain completely knotted structures that are easily detectable by computational models.[21] Mathematically, slipknots are difficult to analyze because they are not recognized in the examination of the complete structure.

A pentafoil knot prepared using dynamic covalent chemistry was synthesized by Ayme et al. in 2012, which at the time was the most complex non-DNA molecular knot prepared to date.[22] Later in 2016, a fully organic pentafoil knot was also reported, including the very first use of a molecular knot to allosterically regulate catalysis.[23] In January 2017, an 8₁₉ knot was synthesized by David Leigh's group, making the 8₁₉ knot the most complex molecular knot synthesized.[24]

An important development in knot theory is allowing for intra-chain contacts within an entangled molecular chain. Circuit topology has recently emerged as a topology framework that formalises the arrangement of contacts as well as chain crossings in a folded linear chain. As a complementary approach, Colin Adams. et al., developed a singular knot theory that is applicable to folded linear chains with intramolecular interactions.[25]

Many synthetic molecular knots have a distinct globular shape and dimensions that make them potential building blocks in nanotechnology.

• Circuit topology of folded linear molecules
• Supramolecular chemistry
• Knotted protein
• Knotted polymers
• Topology (chemistry)
• Knot theory