Molecular Knots

Abstract

The first synthetic molecular trefoil knot was prepared in the late 1980s. However, it is only in the last few years that more complex small-molecule knot topologies have been realized through chemical synthesis. The steric restrictions imposed on molecular strands by knotting can impart significant physical and chemical properties, including chirality, strong and selective ion binding, and catalytic activity. As the number and complexity of accessible molecular knot topologies increases, it will become increasingly useful for chemists to adopt the knot terminology employed by other disciplines. Here we give an overview of synthetic strategies towards molecular knots and outline the principles of knot, braid, and tangle theory appropriate to chemistry and molecular structure.

Keywords: interlocked molecules; molecular knots; supramolecular chemistry; template synthesis; topology.

Figure 1

The impact of knotting on technology: a) Spherical stones thought to be bola weights, which would need to be tied together for hunting, date back 500 000 years.1 b) The oldest surviving man‐made knots are those of the Antrea net, a fishing net made of willow with a 6 cm mesh dating to 8300 BC.2 c) Knots, in the form of quipu, have been used to record and communicate information, with the earliest examples possibly predating the invention of the written word.3 Image (a) “Stone ball from a set of Paleolithic bolas” reproduced from https://goo.gl/vyAh85 (downloaded 5 May 2017) under a wikimedia creative commons license. Image (b) “Pieces of the Antrea net” reproduced from https://goo.gl/y0026E (downloaded 5 May 2017) under a wikimedia creative commons license. Image (c) “Quipu from the Inca Empire” reproduced from https://goo.gl/tqZyPW (downloaded 5 May 2017) under a wikimedia creative commons license.

Figure 2

Knotting exploited by animals: a) Wattana the orangutan tying a knot.4 b) The African weaver bird uses knots to tie its nest securely.5, 6 c) Hagfish knot their bodies to generate force when pulling flesh off a carcass.7 Image (a) reproduced from Ref, 4 with permission from the University of Chicago Press. Image (b) “southern masked weaver by wim de groot” reproduced from https://goo.gl/ZpD09h (downloaded 5 May 2017) under a wikimedia creative commons license. Image (c) reproduced from Ref. 7 with permission from Springer Nature.

Figure 3

Reduced diagrams and writhe: a) Definition of a negative and a positive crossing. b) A knot can be oriented by following its loop in an arbitrary direction. In the depicted orientation, the shown trefoil knot has a writhe (Wr) of 3, as all three crossings are positive. The orientation can be reversed by rotation along the indicated C₂‐axis. c) In a trefoil knot with nugatory crossings, the writhe can take any value. Knot diagrams without nugatory crossings are referred to as “reduced”.

Figure 4

Reidemeister moves and examples of their (supra)molecular equivalents. Reidemeister moves allow transitions between any two diagrams (i.e. conformations) of the same knot or link. They are named after the number of components involved in the movement. a) Reidemeister I refers to the creation or removal of a nugatory crossing, the number of crossings/writhe changes by ±1. It is equivalent to twisting a macrocycle to form an additional loop.24 b) Reidemeister II passes one string over another, the number of crossings changes by ±2 but the writhe remains the same. It is equivalent to moving one molecular strand over another.25 c) Reidemeister III refers to passing a string over a crossing. The number of crossings and writhe are unchanged. Note the nugatory crossing in the left‐hand triketone trefoil knot structure, which is necessary for a Reidemeister III move for any alternating knot.26

Figure 5

Reidemeister moves converting different representations of a trefoil knot into each other: a) A trefoil knot is converted from a D ₂‐symmetric double helix to a D ₃‐symmetric form. b) The same process (corresponding to conformational changes in a molecular structure) applied to Sauvage's trefoil knot27 (in this case molecular D ₃‐symmetry cannot be achieved as one loop is chemically different to the other two).

Figure 6

Chirality in knots. a) The trefoil knot 3₁ is topologically chiral, as it cannot be deformed to its mirror image form 3₁* without the strand passing through itself. b) The square knot 3₁#3₁*, a composite knot obtained by connecting two trefoil knots of opposite handedness, is achiral, as the mirror plane σ projects it onto itself. c) The figure‐eight knot 4₁ can be transformed into its mirror image. By flipping the part shown in red over the part shown in blue, an upside‐down version of the mirror image is obtained after deformation, thereby making it topologically achiral. For the sake of brevity, not every Reidemeister move is shown here for this transformation.

Figure 7

Geometric representations of a topologically achiral knot. a) At first sight the reduced representation of the 4₁ knot does not look achiral. b) However, the introduction of nugatory crossings enables the 4₁ knot to adopt an achiral conformation possessing an S ₄ axis. c) A coordination complex with the 4₁ knot topology (see Scheme 15).33

Figure 8

Alternating and non‐alternating knots. a) 8₁₈ is an alternating knot, as overpasses (red) and underpasses (green) alternate (over‐under‐over‐under etc) around the entire length of the strand, as exemplified along the gray path from A to B. b) 8₁₉ is a non‐alternating knot (over‐over‐under‐under etc), shown on the gray path from A to B. The 8₁₉ knot cannot be represented by a solely alternating crossing pattern.

Figure 9

Torus knots. A torus knot T(p,q) runs p times in the poloidal direction (i.e. through the cavity) and q times in the toroidal direction (i.e. around the cavity) around the surface of a torus without the strand intersecting. Swapping p and q results in the same knot. a) T(2,3) is the trefoil knot 3₁. b) T(3,2) is also the trefoil knot.

Figure 10

Generation patterns for torus, twist, clasp, and pretzel knots. a) For odd values of m, a torus knot X ₁ is obtained (X is the number of crossings). These knots are generated by twisting two strands around each other. For even values of m, two component links are created. b) This construct gives a twist knot for any positive n. Twist knots with an even number of crossings are denoted as X ₁, those with an odd number as X ₂ (in this case X₁ is a torus knot). c) Twist knots are a type of clasp knot C(p,o). Twist knots are C(2,o) clasp knots. d) The generation pattern for (q,r,s) pretzel knots. Pretzel knots consist of left or right‐handed helices connected together (see Section 2.11).

Figure 11

Knot table of all prime knots having up to eight crossings including the unknot 0₁. Torus knots are depicted in red, achiral knots in black, non‐invertible knots in white, and non‐alternating knots in green.

Figure 12

Braid representations of knots. a) A braid for the generation of three‐strand torus knots. A knot is generated from the corresponding braid by connecting opposite ends without generating additional crossings. b) This braid generates torus knots with one additional toroidal revolution for each extra value of n. Following this pattern with additional strands in the braid, any torus knot can be obtained. c) A braid for the generation of a family of achiral knots. For any number of n, a reverse rotated palindrome (RRP) is obtained, indicated by the inversion center i. d) This braid also generates achiral knots for any number n not divisible by 3.

Figure 13

Tangle representations. a) A section of a knot (indicated by a dashed circle) can be split into tangles. The fixed entry points of the string are named after the cardinal directions. b) Some basic tangles for the construction of more complex tangles. c) For the construction of a rational tangle, the starting tangle is reflected along the NW‐SE axis and the new tangle added to the NE and SE crossing points. The resulting tangle can be further extended by the same procedure. d) Generalization of tangle multiplication. T₂ is not restricted to being an integer tangle such as in (c). e) A different way to connect tangles is by addition. The sum is formed by connecting the NE and SE crossing point of the first tangle T₁ to the NW and SW crossing point of the second tangle T₂, respectively.

Figure 14

Unknotting numbers. a) Any twist knot has an unknotting number of 1, as inverting one crossing is sufficient to give the unknot 0₁. b) The unknotting number of a torus knot T(p,q) is $1/2$ (p−1)(q−1), In this example, the pentafoil knot 5₁ is converted into a trefoil knot 3₁ by inverting one of its crossings. Changing a second crossing gives the unknot 0₁. So the unknotting number is 2 (=$1/2$ (2−1)(5−1)).

Figure 15

Changing entanglement using k‐moves. a) A k‐move introduces k‐positive crossings in a set of two strings, a −k‐move introduces k‐negative crossings. b) A supramolecular 3‐move induced by Cu^I ions forms the scaffold for the synthesis of a molecular trefoil knot.27

Scheme 1

Sauvage's synthesis of a [2]catenane (Cu^I 3) by passive62 metal‐template synthesis. The phenanthroline‐Cu^I system formed the basis for the synthesis of several other mechanically interlocked molecular types (rotaxanes, trefoil knot, Solomon link).61 All of the cap‐and‐stick structures shown in this Review are X‐ray crystal structures produced from coordinates taken from the Cambridge Structural Database (CSD).

Figure 16

The linear helicate approach to simple knots and links. Metal ions induce the twisting of the ligand strands to form a double helix (k‐moves, Section 2.12). If the number of crossings is odd, a molecular knot is created upon connecting a/a’ and b/b’. For an even number of crossings, links ([2]catenanes) are produced. The linear helicate approach was successfully demonstrated by Sauvage for the first three in this series (Hopf link, trefoil knot, and Solomon link), but fails for higher order topologies such as the pentafoil knot 5₁ (Section 3.3.1).

Scheme 2

Molecular trefoil knots prepared from a linear helicate strategy. a) Synthesis of trefoil knot Cu₂ 5 a after covalent capture of linear double helicate Cu₂ 4 a ₂ by Williamson ether synthesis.64 b) The yield of the trefoil knot was significantly increased by using RCM for the macrocyclization reactions.65 c) X‐ray structure of the related trefoil knot Cu₂ 5 c. This early design was obtained in lower yield, as the alkyl chain connecting the phenanthroline units is less preorganized than the m‐phenylene unit used in later designs.63 d) A related approach using Fe^II and terpyridine derivatives instead of Cu^I and phenanthroline ligands.66 e) Enantioselective synthesis of a trefoil knot by the linear helicate approach.68

Figure 17

Sokolov's proposed route for the synthesis of a molecular trefoil knot templated by the octahedral coordination sphere of a transition metal. Modified from Ref. 69 with permission from the Royal Society of Chemistry.

Scheme 3

Hunter's synthesis of a trefoil knot using a single metal ion template, via open knot Zn9. Functionalizing the ends of the open knot with alkene units enabled closure to the trefoil knot by RCM. The structure of the open knot was determined by X‐ray crystallography.70, 71

Scheme 4

Synthesis of a trefoil knot by the circular helicate approach. a) A single lanthanide ion entwines three 2,6‐diamidopyridyl ligand strands 14 in its coordination sphere, thereby forming a trefoil knot upon connection of the ligand end groups. The achiral precursor 14 a yields a racemic mixture of the two enantiomers of trefoil knot Ln15 a. The use of C ₂‐symmetric ligand 14 b gives enantiopure trefoil knot Ln15 b. b) X‐ray crystal structure of enantiopure trefoil knot Ln15 b.72, 73

Scheme 5

Synthesis of an enantiopure knot from single ligand strand 16. The addition of Ln salts generates an open knot complex of single handedness which can be closed to the trefoil knot 17.74

Scheme 6

Active template synthesis of trefoil knot 19. One crossing is generated by Cu^I coordination to the bipyridine groups, which forms a loop. A CuAAC reaction of the azide and alkyne termini is directed through the loop by the second coordinated Cu^I ion, thereby forming the trefoil knot.78

Scheme 7

The condensation product of 20 and 21 was originally proposed by Hunter to be [2]catenane 23.81 Several years later, X‐ray crystallography by the Vögtle group showed that the product was actually trefoil knot 22.82 A network of hydrogen bonds responsible for directing the assembly of the knot is shown by dashed lines.

Scheme 8

A trefoil knot 25 obtained by ring closure of steroid trimer 24. An extended network of hydrogen bonds is visible in the solid‐state structure, which has almost perfect C ₃‐symmetry (the hydrogen bonds are indicated by dashed black bonds in the crystal structure and as red dashed lines in one of the subunits in the diagram).84

Scheme 9

A molecular trefoil knot 27 discovered in a dynamic covalent library. The interlocked structure minimizes the exposure of the hydrophobic surface area to the solvent by burying part of the molecule in the central cavity.87

Scheme 10

Synthesis of trefoil knot 28 based on imine exchange.88 During the crystallization process, two bromide anions are incorporated in the central cavity, one above the other.90 Attempts at crystallization in the absence of Br⁻ were unsuccessful.

Scheme 11

a) Schill's approach towards a molecular trefoil knot using a covalent scaffold. Trimerization of a crowded quinone 31 was projected to give a molecular trefoil knot after cyclization and hydrolysis.92 b) Walba's Möbius strip approach towards molecular knots. Introducing a half twist in compound 32 before connecting the ends yields molecular Möbius strip 33. Statistical twisting was too disfavored to yield trefoil knot 34 after ozonolysis.93

Scheme 12

Towards the synthesis of molecular knots by using covalent scaffolds. a) A hybrid approach towards a molecular trefoil knot by using Cu^I and a 1,3,5‐substituted benzene as the template. The interlocked structure of 35 after cyclization was confirmed by X‐ray data, but it was not possible to remove the central benzene template to yield a knot.96 b) A trefoil knot assembled around a benzene‐1,3,5‐tricarboxylic acid template. The template could be removed after cyclization, but the formation of 36 could not be confirmed experimentally.26

Scheme 13

Synthesis of a metalla‐trefoil knot by trimerization of ethylene glycol bridged quinolines 37 with Ag^I. Coordination bonds from oxygen to silver are omitted for clarity.33

Scheme 14

Synthesis of molecular figure‐eight knot 39 by disulfide exchange in a dynamic covalent library. Knotting likely results from the hydrophobic effect.99

Scheme 15

Synthesis of a figure‐eight metalla‐knot by tetramerization of ethylene glycol bridged quinolines 40 with Ag^I. Coordination bonds from oxygen to silver are omitted for clarity.33

Scheme 16

Synthesis of a linear double helicate with five crossings. Attempts to ring‐close Li₄ 41 ₂ to the corresponding pentafoil knot (5₁) were unsuccessful.97

Figure 18

Incorrect registry of ligands disfavors formation of a desired topology. In addition to an increased probability of connecting the wrong strand ends of a linear helicate, partly interwoven helices can become kinetically trapped with longer ligands.

Figure 19

Transition from a linear double helicate to a circular double helicate. a) One of the limiting factors of the linear helicate approach is the increasing distance between the ends. b) Bending the helix brings the ends closer together, but does not reduce the length (and complexity) of the ligand strands. c) In a circular helicate, an additional metal ion brings additional organization to the ends of the helix. The symmetry enables shorter (simpler) ligands to be used at the cost of requiring more reactions (five as opposed to two for the linear helicate in the case of a 5₁ knot) to achieve closure of the loop.

Scheme 17

Synthesis of molecular pentafoil knots (5₁) via circular double helicates. a) Fe₅ 44 is obtained by formation of an imine bond between ligand 42 and diamine 43 in the presence of Fe^II. The size (pentamer) of the circular helicate is determined by a chloride anion template.105 b) The yield of the pentafoil knot is increased by using ligand 45, which allows for covalent capture of the closed‐loop knot by RCM. In contrast to Fe₅ 44, Fe₅ 46 does not decompose upon demetalation.107 c) Solid‐state structure of Fe₅ 46 (the structure of Fe₅ 44 was also determined by X‐ray crystallography).

Figure 20

The complexity of knots accessible from helicates increases from a) a linear double helicate to b) a circular double helicate to c) a circular triple helicate.110

Scheme 18

Synthesis of an 8₁₉ knot based on the circular triple helicate approach. Tetramer Fe₄ 47 ₄ is formed by the reaction of ligand 47 with FeCl₂. Covalent capture by RCM yields knot Fe₄ 48.110.

Figure 21

Synthesizing composite knots by forming the knot sum. a) Forming the homodimer of a chiral knot can result in two enantiomeric knots of opposing handedness (if two knots of identical chirality are joined) and one achiral knot (by joining knots of opposing handedness). This is analogous to forming dimers of a racemic compound: a meso diastereomer (combining R and S) is obtained as well as chiral diastereomers (combining R and R or S and S). b) Forming the knot sum of two different chiral knots gives four distinguishable knots, analogous to joining two different chiral centers in a molecule.

Scheme 19

Synthesis of a mixture of molecular composite knots. The dimerization of two open trefoil knots 49 leads to the formation of molecular granny knot 50 and molecular square knot 51 among other products, as deduced by MS and NMR data. Which knot is formed is determined by the handedness of the two open trefoil precursors. If two complexes of the same handedness are combined, a granny knot is obtained, while the combination of two complexes of opposing handedness yields a square knot.113

Figure 22

Circular dichroism (CD) spectra of the two enantiomers of a) Sauvage's trefoil knot 5 a 67 and b) pentafoil knot 46.107 Neither knot has elements of Euclidean chirality. Reproduced from Refs. 67 and 107 with permission from Wiley‐VCH and the American Association for the Advancement of Science, respectively.

Figure 23

Asymmetric catalysis of the Mukaiyama aldol reaction by a chiral trefoil knot.74

Figure 24

Allosteric regulation of Lewis acid carbocation catalysis by a molecular pentafoil knot. a) Remetalation of the knot 46 with Zn²⁺ gives the metalated knot with an empty central cavity. b) The metalated knot can remove bromide from trityl bromide to give the catalytically active trityl cation. c) Subsequent demetalation regenerates the organic knot ligand 46, thereby shutting down the catalytic activity.107

Figure 25

Knotted and interlocked cyclic polymers imaged by AFM.125 Adapted from Ref. 125 with permission from Wiley‐VCH.

Figure 26

An actin filament tied in a knot by molecular tweezers.129 Tightening of the knot eventually causes the filament to break at the location of the knot. Adapted from Ref. 129 with permission from Springer Nature.

Figure 27

Knotted DNA (with conventional representations of knots) produced on incubation of circular DNA with a topoisomerase I enzyme and imaged by electron microscopy.¹³⁶ Adapted from Ref. 136 with permission from the American Society for Biochemistry and Molecular Biology.

Figure 28

Schematic representation of a loosely knotted protein.154 A trefoil knot is formed by passage of the B9 β‐strand leading to the C‐terminus through a loop formed by the sequence B1→[central domain]→B5→H3→B6. Adapted from Ref. 154 with permission from the American Chemical Society.

Figure 29

X‐ray crystal structure and reduced schematic diagram of DehI, the most complicated knotted protein known to date, which contains a Stevedore knot (6₁).156 Adapted from Ref. 156 with permission from PLOS.