Molecular knots in biology and chemistry

A DNA knot is defined as the self-entanglement of a single DNA molecule, therefore this excludes catenane structures that are formed by more than one chain (figure 5(a)). In 1976, Liu and co-workers first discovered that single-stranded DNA chains in bacteriophages could knot when treated with Escherichia coli omega protein, a type I topoisomerase [30]. This was subsequently followed by the discovery of knots in double-stranded DNA chains in 1980 when a supercoiled plasmid was incubated with excess amounts of type II topoisomerase from bacteriophage T4 [31]. Since then, various knotted structures formed in nicked, circular duplex DNA molecules by E. coli topoisomerase I have been identified in vitro, ranging from simple trefoil knots to more complex higher order and composite knots (figure 5(b)) [9]. With the use of electron microscopy imaging and agarose gel electrophoresis, Dean and co-workers characterised these topologically different knotted DNA structures in detail [9].

Zoom In Zoom Out Reset image size

In the last three decades, an increasing number of studies of DNA knots have been undertaken [32–35]. As discussed above, knots in DNA can form in vitro when DNA strands are cut and re-joined with the help of topoisomerases. DNA topoisomerases control the topology of DNA by introducing transient breaks in DNA strands then re-ligating them to different ends [36, 37]. They are classified into two types: type I or type II. Type I topoisomerases mediate the passage of a single strand of duplex DNA through a nick in the complementary strand. In contrast, type II topoisomerases introduce a transient double-stranded break in one segment of the DNA, allowing a second segment of duplex DNA to pass through before the strands are chemically ligated. A variety of knotted DNA products can also form when recombinases act on supercoiled circular DNA substrates (an example is shown in figure 5(c)) [38–40]. Recombinases are involved in changing the topology of DNA by a complex process called site-specific recombination [41]. In this case, they mediate genome rearrangement such that a DNA segment is inserted, excised or inverted in accordance with the appropriate recombination sites [41].

DNA knots can also arise in vivo during replication and transcription, as these processes require the action of topoisomerases to release accumulated torsional stress in the DNA [42]. In partially replicated bacterial plasmids with two origins of replication in head-to-head orientation, it has been observed that topoisomerases induce knot formation within replication bubbles that are helically wound (figure 5(d)) [35]. Olavarrieta and co-workers have also shown that complex knotting of the duplex DNA in small pBR322-derived plasmids can be initiated by a head-on collision of replication and transcription, resulting in plasmid instability in E. coli (figure 5(e)) [43]. Recently, the Schvartzman group has suggested that if the progression of the replication forks in DNA synthesis is impaired, sister duplexes can become loosely intertwined and this can lead to the introduction of knots by the action of topoisomerase IV (Topo IV) [44]. It should be noted, however, that these observations are made on small bacterial plasmids and whether they are applicable to large bacterial or eukaryotic chromosomes is still uncertain.

Several studies have also previously reported that linear viral genomic DNA can cyclise and form knots upon extraction from P4 bacteriophages (figure 5(f)) [31, 45]. Furthermore, it was found that the probability of DNA knotting was enhanced in intact P4 deletion mutants [46] and tailless P4 phages [47]. In a series of experiments, Arsuaga and co-workers showed that most viral DNA molecules (>95%) are highly knotted due to the tight confinement and writhe bias of their packing geometry within the phage capsid (figure 5(g)) [7, 33]. Writhe is the amount a piece of DNA is deformed to form coils as a result of torsional stress, which leads to the phenomenon of DNA supercoiling. Although the specific mechanism of knot formation is still unclear, characterisation of the complex knot spectrum of bacteriophage P4 genome by high-resolution gel electrophoresis revealed that chiral and torus knots were favoured by confinement over achiral and twist ones [7]. Results from recent simulations also showed that there was a preference for chiral knots, albeit no significant bias of torus over twist knots was found [48]. As yet, it remains to be seen what factors actually determine viral genome organisation in terms of its knot types and distribution.

How does DNA knotting affect its biological activity within cells? As discussed above, several processes such as DNA compaction, topoisomerisation, site-specific recombination, replication and transcription can result in the formation of DNA knots in cells. However, the presence of knots in DNA has potentially detrimental effects in several cellular processes such as transcription and replication [50–52] and, if unresolved, can lead to mutational defects in the genome or even cell death. To overcome these problems, cells express and produce essential, ubiquitous enzymes called topoisomerases, which can remove knots promptly and efficiently [53, 54]. Contrary to this, it has to be noted that these enzymes also play a role in creating DNA knots. As a result of their presence and dual-functionality, cells have evolved and taken advantage of the topologically constrained nature of their DNA. Lopez and co-workers demonstrated that Topo IV in bacteria can not only form knots in DNA during replication but it is also responsible in unknotting them later on so that DNA can get correctly segregated to every daughter cell [44].

In the case of bacteriophages, recent simulations have revealed that the organization and topology of packaged DNA in capsids are important in how fast the DNA gets ejected into an infected bacterial cell [55]. Marenduzzo and co-workers observed that ordered DNA spools in the capsid, favoured by DNA cholesteric interactions, were ejected at a faster rate than disordered, entangled DNA [55]. It was also shown that torus knots exited the capsid more easily than twist knots, which can halt the ejection process.

DNA is an extremely long biological polymer, and it is no surprise therefore that linear and circular, single- and double-stranded DNA molecules are all known to form a wide range of knotted structures from simple trefoil (3₁) knots to more complex knots such as those with nine crossings. Whereas there are examples of DNA forming both chiral and achiral knots as well as torus and twist knots, there is some evidence, at least in the context of highly packaged viral genomic DNA, that there is a preference for chiral and torus knots. In many cases, it is well established that DNA becomes knotted as a direct result of biological processes such as recombination, replication and transcription. In these cases, knotting is problematic and, consequently, numerous enzymes exist (topoisomerases) which catalyse the unknotting of a DNA chain through a 'cut and paste' mechanism in which the DNA is first cut, then moved/rotated and subsequently religated. Effectively, this breaks the chain into small segments and rearranges them to eliminate the knot. The biological consequences of not removing the knot can be severe, e.g. cell death. In contrast, there may also be benefits of knotting, such as the case of highly packaged viral genomes. Here, knotting may aid in the tight packing and it can also affect the rate at which the genomic DNA is ejected from its viral carrier/storage compartment, the capsid.

DNA can also form a range of other topologically complex states including catenane structures such as Hopf and higher-order links. For a comprehensive overview on the various topological forms of DNA, interested readers are directed towards the following references [8, 37, 56–58].

In addition to all of the studies discussed above on knotting in naturally occurring DNA, there is also considerable literature on knotting in synthetic single-stranded DNA. In particular, Seeman and co-workers have been able to rationally design and build synthetic forms of DNA with a range of knot types and links. A detailed discussion of this work is out of the scope of this review, however, a summary of the different types of structures that have been synthesised is given in table 3, and interested readers are directed to the references provided in the Table.

A pseudoknot is generally defined as an RNA structure that consists of at least two helical segments linked together by single-stranded regions or loops [62]. Although pseudoknots can possess several distinct folding topologies, the best characterised to date is the so-called H (hairpin)-type or classical pseudoknot. As illustrated in figure 6, this is the simplest type of pseudoknot structure that results from the base pairing of a single-stranded segment of RNA in the loop of a hairpin to a complementary sequence outside the loop region. It comprises of two base-paired stem segments (S1 and S2) and, depending on the number of loop bases involved in the pseudoknotting interaction, two or three single-stranded connecting loops (L1, L2 and L3) [63]. However, in most classical pseudoknots (>85% [64]), L2 is missing and thus S1 and S2 can coaxially stack on top of each other to form a quasi-continuous helix. Figure 6(d) depicts this arrangement in the H-type pseudoknot structure of the 3'-terminus of the TYMV RNA, where L1 spans S2 and crosses the deep groove of the helix whilst L3 spans stem S1 and crosses the minor groove. In addition to coaxial stacking, pseudoknots can also be further stabilised by hydrogen bonds formed between single-stranded loop regions and the adjacent stem segments. As the connecting loops and stems can vary in length, and the interactions between them can differ, RNA pseudoknots represent a structurally diverse group. Hence, it comes as no surprise that these structures are associated with various vital roles in biology. These include forming functional domains within ribozymes [65] and telomerase [66] as well as inducing ribosomal frameshifting in many viruses [10, 67] and regulating translation [68].

Zoom In Zoom Out Reset image size

The RNA pseudoknot is a ubiquitous folding topology that has been identified in almost all organisms [14]. Below, we describe well-characterised examples of pseudoknots involved in catalysis, ribosomal frameshifting and translational regulation, highlighting how the structures are related to their function. In most cases, it has also been shown that the function of pseudoknots is associated with their position along the RNA sequence [63, 69, 70]. For example, pseudoknots located at the core of the tertiary fold of RNAs tend to be crucial in catalysis whilst those found at the 5' end of mRNAs are typically involved in translational control. In addition, in non-coding regions (NCRs) of viral RNAs, pseudoknots play a role in the regulation of initiation of protein synthesis and in template recognition by viral replicases.

4.2.1. Catalytically active pseudoknots.

Catalytic RNAs, or ribozymes, are RNA molecules that can catalyse specific biochemical reactions. It has been shown that most ribozymes fold into similar 3D structures that are essential for their function [71]. As a model to understand the mechanism of catalytic RNAs, extensive studies have been done on the hepatitis delta virus (HDV) ribozyme, the fastest known naturally occurring self-cleaving ribozyme [72–74]. HDV is a satellite RNA virus of hepatitis B virus, which together can cause severe infection in humans [75]. The host RNA polymerase II replicates the circular genome of HDV through a double rolling-circle mechanism, producing long RNA transcripts that must be cleaved for viral replication. The processing of the HDV RNA is performed by the self-cleaving HDV ribozyme encoded in the RNA [76]. As illustrated in figure 7(a), the HDV ribozyme has a characteristic 'nested' double pseudoknot that not only forms the active site necessary for the specificity of substrate binding and catalysis but also stabilises the overall RNA structure [77]. This pseudoknot motif has also been discovered in other small self-cleaving ribozymes, particularly in the core of glmS ribozymes in many Gram-positive bacteria [78, 79] and mammalian cytoplasmic polyadenylation element-binding protein 3 ribozymes [80]. As a result, these RNAs are able to achieve an overall complex and stable conformation.

Zoom In Zoom Out Reset image size

Eukaryotic chromosomes possess telomere ends that protect themselves from loss of genetic material due to successive DNA replication events [81]. Maintenance of the telomeres is performed by the ribonucleoprotein telomerase, an RNA-dependent DNA polymerase made up of a specialised reverse transcriptase and a telomerase RNA (TR) [82, 83]. Although telomerase activity is essential for highly proliferative cells such as stem cells, it is also known to be elevated in ~90% of cancer cells [84, 85] and may play a role in aging [86]. TRs not only provide the template for DNA synthesis but also contain a highly conserved classic H-type pseudoknot within the core domain, which is needed for telomerase assembly and activity [87–90]. Figure 7(b) shows a structure of the human TR pseudoknot, where triple nucleotide interactions U—A-U between L1 and S2 in the deep groove form a triple helix important for telomerase repeat addition processivity [66]. Studies have also shown that the conformational switch that exists between the pseudoknot and a less stable hairpin might be crucial for telomerase activity [91, 92]. Mutations in the TR pseudoknot have also been associated with inherited human disorders such as aplastic anemia and autosomal dyskeratosis congenital [86, 93, 94].

4.2.2. Ribosomal frameshift-inducing pseudoknots.

4.2.3. Pseudoknots involved in translational regulation.

The function of an RNA molecule can often be inferred from its 3D structure. Since RNA structures are hierarchical and the structural determination of their 3D conformation using experimental methods is difficult, RNA secondary structure prediction is important in elucidating the potential structures and therefore, functions of RNAs. A number of different approaches to RNA pseudoknot structure prediction have been developed over the last decade. These are described below.

Most pseudoknot-free structure prediction programs are based on determining a minimum free-energy (MFE) conformation from the primary nucleotide sequence. However, the prediction of RNA pseudoknots is computationally complex as the search for a MFE structure, in these cases, has been shown to be a Non-deterministic Polynomial-time (NP)-complete problem with respect to sequence length [113]. Dynamic programming (DP)-based methods, which use free energy minimization, can only predict limited classes of pseudoknots. For example, in the case of PKNOTS, the algorithm accurately predicts structures for RNA sequences of length up to 100 bases [114]. Other programs that also use the DP-method include NUPACK [115] and pknotsRG [116]. These approaches, however, are effective only for short sequences, as computation time can increase as the third to sixth power of sequence length, depending on the algorithm used [114, 115, 117].

To overcome this issue, heuristic prediction methods such as FlexStem [118] and HotKnots V2.0 [119] have been developed. Although the predicted structure is not necessarily the MFE, such approaches can handle a wider class of pseudoknots and longer sequences. In another case, the IPknot method, developed by Sato and co-workers, can predict pseudoknotted structures from sequences up to 1000 bases with increased speed and accuracy [120]. Based on integer programming (IP), this method breaks down the pseudoknotted structure into pseudoknot-free substructures and approximates a base-pairing probability distribution that considers pseudoknots. In addition, it can also use multiple aligned sequences to predict a consensus pseudoknotted structure [120].

Another algorithm that can predict the MFE RNA pseudoknot structure is TT2NE, which is based on classifying RNA structures according to their genus [121]. Although it can only predict structures for sequences up to 200 bases, it has been shown that the quality of predictions is significantly improved when compared to other state-of-the-art algorithms [121]. Based on the same concept, the same group recently developed McGenus, a Monte Carlo algorithm [122]. Here, the method stochastically searches the MFE structures from sequences of up to 1000 bases. More recently, Jabbari and co-workers have developed an iterative-based method called Iterative HFold, which uses a pseudoknot-free structure to predict pseudoknotted structures rather than a sequence as input [123].

Pseudoknotted structure prediction programs are a valuable resource; examples of some of these recent programs and webservers are listed in table 1. Further details of currently available pseudoknot structure prediction programs can be found elsewhere [124–126]. In general, most of the approaches have been developed with the aim of predicting pseudoknotted structures with increased speed and accuracy. However, it remains clear that these algorithms are still restricted by the lack of understanding of pseudoknot thermodynamics and the capacity to cope with pseudoknots containing stem regions with bulged residues or non-Watson-Crick pairs. In addition, steric constraints and the contribution of entropy to the free energy are often ignored, as there is limited information on the full 3D geometry of pseudoknots. Environmental factors such as ions, solvent, protein and other RNAs are also important in the structure and function of RNA; and ideally these also need to be accurately incorporated into the predictions.

In contrast to DNA, naturally occurring RNA, strictly speaking, does not form knotted structures. However, it frequently adopts structurally complex conformations in which there are a number of topological crossings of its chain. These structures are known as pseudoknots and are widespread in terms of the different classes of RNA in which they are found. They vary in the length and presence/absence of loop regions and therefore represent a structurally diverse group. It is perhaps, therefore unsurprising that the pseudoknot structure is associated with a range of different biological processes, including catalysis, ribosomal frameshifting and regulation of translation. Although it is not completely understood how their structure results in their specific activities, it is clear that the pseudoknot structure is stable (although it can be in equilibrium with other conformations such as hairpins), may be particularly stable with respect to unwinding by helicases, or degradation. Prediction of the structure of pseudoknots in RNA has rapidly developed over recent years, and, although it is still challenging for very long sequences, a number of different approaches can be used which are increasing in speed and accuracy. Interested readers are directed towards the following references for a more detailed discussion of all of these topics [10, 11, 14, 63, 68, 69]. It is interesting to note that RNA sequences have been designed to form a synthetic trefoil knot [132], see Discussion for further details.

For a long time, it was thought that it was highly unlikely, if not impossible, for a polypeptide chain to 'knot' itself to form a functional folded protein. This was, in part, due to the fact that, at that time, no examples of deeply knotted proteins were identified within the protein data bank (PDB) [147]. In this study, a very shallow knot was discovered in carbonic anhydrase by Mansfield [147]. One of the challenges in the search for protein knots was the difficulty in determining whether a knot is present within a complex structure. Thus, for many years, knots in protein structures went undetected. As various computational and mathematical tools were developed to detect and identify knots, it became clear that topologically knotted protein structures do exist, even some with extremely deep knots [24, 26, 148, 149]. Now there are a few web-servers that have simplified the task of knot identification in proteins and can determine quickly whether a structure contains a knot and, if so, what type [150, 151]. In addition, the recent KnotProt database (http://knotprot.cent.uw.edu.pl/) created by Sulkowska and co-workers classifies knotted proteins and represents their knotting complexity (knot type and depth of knot) as a 'knotting fingerprint' in the form of a matrix diagram [142, 152, 153]. Matrix diagrams, which are an excellent method for visualising knots and slipknots in proteins, were originally used in the analysis of slipknots in proteins by the Yeates group [140].

To date, over 750 knotted proteins have been discovered within the PDB, equivalent to approximately 1% of all entries [152]. A current list of examples of these structures is provided in table 2. It is worth noting that the KnotProt database is updated regularly [152]. Over the years, a growing number of knotted proteins have been observed in all three domains of life [15, 142, 154]. These include structures that contain a trefoil (3₁), figure-of-eight (4₁), Gordian (5₂) and stevedore (6₁) knot with three, four, five and six projected crossings of the polypeptide backbone, respectively (figure 9).

Zoom In Zoom Out Reset image size

Trefoil knots are the most prevalent and simplest type of knot discovered in proteins. The first protein trefoil knot to be identified was that found in carbonic anhydrase—a family of proteins involved in catalysing the reaction of carbon dioxide to hydrogen carbonate and H⁺ [147]. This trefoil, however, is rather shallow as the C-terminus extends through a wide loop by only a few residues. A few years after Mansfield's 1994 study, a much deeper trefoil knot was detected in E. coli S-adenosylmethionine synthetase, an enzyme that catalyses the reaction between methionine and ATP [155, 156]. By far, the largest and most well-studied family of deeply knotted proteins is the trefoil α/β knot fold—a class of methyltransferases (MTases) which are members of the SpoU family [157, 158]. These knotted proteins share common structural features and it is highly likely that all are MTases that catalyse the transfer of the methyl group of S-adenosyl methionine (AdoMet) to carbon, nitrogen or oxygen atoms in DNA, RNA, proteins and other small molecules [159]. In solution, all form dimers with the knotted region comprising part of the AdoMet binding site and forming a large part of the dimer interface [157, 160–163]. Trefoil knots have also been found in two homologues of N-succinylornithine transcarbamylase; the AOTCase from X. campestris catalyses the reaction from N-acetylornithine and carbamyl phosphate to acetylcitrulline [164], and SOTCase from B. fragilis promotes the carbamylation of N-succinylornithine [165]. Besides being found in enzymes, trefoil knots have also been identified in Rds3p, a eukaryotic metal-binding protein essential for pre-mRNA splicing [166] and more recently, in the family of sodium/calcium exchanger membrane proteins [152].

More complex knots have also been identified in proteins that catalyse various enzymatic reactions. A deeply embedded, figure-of-eight protein knot has been found in plant ketol-acid reductoisomerases, which are involved in the biosynthesis of branched-chain amino acids [167, 168]. In addition, a Gordian knot has been identified in the family of mammalian ubiquitin carboxyl-terminal hydrolases (UCHs); the proteins are deubiquitinating enzymes that catalyse the cleavage of the isopeptide bond formed between ubiquitin and lysine side chains of protein and other adducts, and thus are involved in the ubiquitin-proteasome system [169–171]. The most complex protein knot known to date is the 6₁ stevedore knot discovered in DehI, a α-haloacid dehalogenase that catalyses the removal of halides from organic haloacids [154]. Apart from these enzymes, it has been shown that the figure-of-eight knot also exists in the chromophore-binding domain of a red/far-red photoreceptor phytochrome from bacterium D. radiodurans [172, 173].

Slipknotted structures have also been found in a number of proteins (figure 8(f)) [140]. They cannot be identified using the standard methods for knot detection in proteins as, in these cases, the knot becomes undone when the chain is pulled at both termini. As such, it comes as no surprise that these structures had been overlooked until relatively recently. In 2007, Yeates and co-workers first discovered a number of protein slipknots by using an approach based on the fact that slipknots become real knots at some point when the polypeptide chains are shortened [140]. At present, over 450 protein slipknots have been identified [152] and a list of examples of these structures is listed in table 2. It is worth noting that the KnotProt database is the first, and currently only, database that provides details on slipknotted structures [152].

Alkaline phosphatase is the largest family of proteins that contain deep slipknots [15, 140, 152]. In the case of E. coli alkaline phosphatase, 30 residues have to be deleted from the C-terminus before a knotted conformation results. Similar to that of knotted proteins, many of the protein slipknots discovered to date are also found in other enzymes such as thymidine kinases and sulfatases [15, 140, 152]. Interestingly, slipknots have also been found in transmembrane proteins that span the entire cell membrane to which they are permanently embedded [15, 140, 152]. Examples include the families of sodium:neurotransmitter transporters, betaine/carnitine/choline transporters (BCCT) and proton:glutamate transporters [142].

Further details of knotted and slipknotted protein structures can be found in other recent reviews [12, 13, 15, 174] and the KnotProt server [152]. It should be noted that the KnotProt database also provides extensive key information about the biological functions of proteins with knots and slipknots [152].

Topologically knotted proteins have been found to be conserved across different families [142], suggesting that the knot itself may be advantageous and important to the function of the protein. It has been speculated that a knotted topology could play a key role in increasing catalytic activity or ligand binding affinity (potentially by decreasing dynamics) or enhancing stability (thermodynamic, kinetic and mechanical) of a protein. As yet, relatively little is known about the functional advantages, if any, of these complex knotted structures over their unknotted counterparts. However, various experimental and computational studies have been undertaken to address this question.

Many reports have shown that the knotted regions of knotted proteins play crucial roles in enzymatic activities and ligand binding. As discussed in section 5.1., it has been observed that the knotted regions of the proteins in the α/β-knotted SpoU MTase family comprise part of the active site to which the ligand binds (two examples of α/β knot MTases are illustrated in figure 10(a)) [159–162]. In the case of the N-succinylornithine transcarbamylase, Virnau and co-workers have demonstrated through a computational study that the presence of the knot in the knotted homologue AOTCase may structurally modify its active site and subsequently, may alter its enzymatic activity (in terms of substrate specificity) compared to its unknotted homologue OTCase (figure 10(b)) [149]. In addition, structural studies of the D. radiodurans phytochrome revealed that the deeply embedded knot in the chromophore-binding domain is in contact with the chromophore [172, 173]. A recent study on the conservation of knotting fingerprints in UCHs also showed that there was a correlation between the locations of active site residues and points characterising its knotted topology (i.e. the knotted core) [142]. Despite these examples, there is still little direct experimental evidence that a knotted structure can influence the activity of a protein.

Zoom In Zoom Out Reset image size

The question of whether knots have any effect on the conformational dynamics of proteins has also been raised. In the phytochrome protein, it has been noted that the figure-of-eight knot sits where increased rigidity could be important in driving conformational changes that occur when light energy is absorbed by the chromophore [172, 175]. Recent computational approaches using simple lattice models have shown a narrow and less extended native basin for a 5₂-knotted structure relative to a similar but unknotted one, suggesting enhanced rigidity [176]. However, experimental studies by Andersson et al, which measured ¹⁵N spin relaxation parameters using NMR experiments for the 5₂-knotted UCH-L1, reported no significant differences between the relaxation properties of the knotted protein relative to unknotted proteins of a similar size [177]. Thus, it remains to be clearly established, particularly experimentally, whether knotted structures can influence the conformational dynamics of a protein.

Much research effort has been undertaken to address the question of whether a knot can provide additional thermodynamic, kinetic or mechanical stability to a protein structure. Sulkowska et al performed coarse-grained simulations of the thermal and mechanical unfolding of the knotted (AOTCase) and unknotted (OTCase) variants of the transcarbamylase-like proteins as well as a synthetic construct of the knotted parent protein rewired so as to remove the knot [178]. In this case, the knotted structure was found to have longer unfolding times than the other two unknotted proteins, which were attributed to topological and geometrical frustration [178]. In an attempt to investigate the potential thermal stabilities of knotted proteins in an experimental study, Yeates and co-workers engineered a knotted and an unknotted ('superficially knotted') polymer [179]. They showed that the knotted chain had a higher thermal stability than the unknotted one (figure 10(c)), although it is important to note that the unfolding in both cases was not fully reversible and therefore only apparent melting temperatures were reported. However, computational studies using Monte Carlo simulations of a simple lattice model using Gō-like potentials showed that a trefoil knot did not have any effect on the thermodynamic stability of a simple protein structure [180]. Instead, it was found that the knot enhances kinetic stability as the knotted protein unfolds at a distinctively slower rate than its unknotted counterpart [180]. Further studies by the same group demonstrated that a more topologically complex protein knot, the 5₂ knot, clearly enhanced the protein's kinetic stability in comparison to that of a protein containing a 3₁ knot [176].

The resistance of knotted proteins to mechanical unfolding has been examined by atomic force microscopy (AFM). The first system to be studied was the shallow trefoil-knotted carbonic anhydrase B. In this particular case, an extremely high resistance to unfolding was observed when the protein was pulled from its termini in contrast to a considerably lower resistance when the molecule was pulled from other positions resulting in the untying of the knot [181, 182]. Although these initial studies suggested a dramatic effect of a knot on mechanical stability, the results have not been observed in AFM studies of other knotted systems [175]. In the case of carbonic anhydrase B, recent simulations have shed light on the possible reasons for its remarkable mechanostability [183]. These studies revealed that after an initial, rather limited unfolding event, the knot is wrapped around an inner β-sheet structure in the core of the protein. Thus, the knot is tightened but effectively locally captured by a structural obstacle in the chain. This is aided by the stabilising effects of a zinc ion, which coordinates to the region that becomes entangled by the knot. The simulations explain why in the AFM experiments, the contour length observed is so much smaller than that expected for a fully stretched polypeptide chain containing a tightened knot. In an interesting extension of their initial work, Ikai and co-workers made a tandem repeat of carbonic anhydrase B. Combining AFM with biochemical measurements of activity and binding, they were able to establish that the C-terminal knotted region was essential for activity [184].

The mechanical stability of the 4₁-knotted phytochrome protein has also been investigated by Bornschögl and co-workers using AFM [175]. In this case, however, they did not observe any enhanced resistance when the knot was tightened as the extension force for unfolding (73 pN) was within the range found for other unknotted proteins. It appears that whether a knot contributes to mechanical stability or not, may depend upon a number of factors including other aspects of the protein's structure and potentially pulling speed/force etc. Several computational studies have suggested that knotting might increase a knotted protein's mechanical stability, thus making it more resistant to cellular translocation and degradation pathways [149, 178, 185, 186]. Again, whether knotting confers any advantageous stabilising effect to a knotted protein over its unknotted counterpart is still inconclusive and thus remains to be tested with more experimental and computational studies.

The significant number of protein slipknots that have now been identified has also posed the question of whether such topologies have any functional or structural role in the protein. In the case of the homodimeric E. coli alkaline phosphatase, Yeates and co-workers engineered cysteine residues at various positions in the protruding loop of the slipknot such that inter-molecular disulphide bonding between the two subunits resulted in a knotted system [140]. Using thermal denaturation, the results showed that the knotted mutants were more thermally stable than either the wild-type or other control mutants. This suggested that the slipknot in the structure may play a role in the enzyme's thermostability [140]. It is also worth noting that the slipknotted B116-like protein is found in a virus that infects thermophilic Sulfolobus archaebacteria [140]. In another study, knotting fingerprint analyses of transmembrane transporting channels from five different families of proteins showed that the slipknotted topology is conserved. This has led to speculations that the slipknot loop, which straps together several transmembrane α-helices, may stabilise their location inside the membrane during their transporter and symporter action [142] (see figure 10(d) for examples of the structures of two slipknotted transmembrane proteins).

The study of how proteins achieve their unique 3D conformation (native state) has been the focus of many researchers in the field of protein folding. For many decades, extensive folding studies focussed on small, monomeric proteins and thus mechanisms of how they fold are now relatively well established [187–191]. These include the framework, nucleation-condensation and hydrophobic collapse mechanisms, which can be viewed as points on a spectrum of a unified mechanism [187, 188]. Current folding theories have shown that small, monomeric proteins, which fold efficiently and rapidly, can achieve their low-energy native configuration from an ensemble of denatured polypeptide chains in a highly cooperative manner and traverse relatively smooth, funneled energy landscapes [192, 193]. However, it is still unclear how these concepts and mechanisms are applicable to larger proteins with more complex topologies including the classes of knotted and slipknotted proteins. Not only do such proteins have to avoid kinetic traps but they also have to overcome significant topological barriers during folding. This section summarises recent developments made towards understanding the mechanisms involved in the formation of these types of complex structures.

5.3.1. Experimental studies on knotted proteins.

Although the elucidation of how knotted proteins fold using experimental approaches remains challenging, in recent years, some significant progress has been made. Most of the experimental folding studies on knotted proteins have focussed on the trefoil-knotted α/β MTases, YibK from H. influenzae and YbeA from E. coli [194–201]. Both proteins are homodimers, which bind to the co-factors AdoMet and S-adenosyl homocysteine (AdoHcy) and contain a trefoil knot at the C-terminus in which at least 40 residues pass through a similarly sized loop (figure 10(a)) [160, 202]. Extensive biophysical techniques have been employed to probe the knotting and folding mechanisms of purified, recombinantly expressed YibK and YbeA. Both unfold reversibly in vitro upon addition of chemical denaturant with a concomitant loss of secondary and tertiary structure [195, 198]. Kinetic studies demonstrated that YibK and YbeA fold similarly via sequential mechanisms that involved one or more monomeric intermediate states and a slow rate-limiting dimerization step [196, 198].

To probe chain knotting events during the folding of YibK and YbeA, Mallam and co-workers constructed a set of knotted fusion proteins in which A. fulgidus This, a stable 91-residue protein, was fused to the N-, C- or both termini of both MTases [201]. This was used as a 'molecular plug' in an attempt to disrupt threading events or to prevent the chain from knotting altogether. Remarkably, these experiments established that both proteins can withstand the fusion of additional domains to both their N- and C-termini and are able to fold to native or native-like states capable of binding cofactor. The fusion proteins created in this study represent some of the most deeply knotted proteins known, the C-terminal fusions requiring some 140 or more residues to pass through a loop to form the knotted native state. Surprisingly, all the fusion proteins showed unfolding and refolding kinetics very similar to the parent MTase giving the first hint that the polypeptide chain might remain knotted even in a highly unstructured chemically denatured state. This was subsequently shown to be the case through in vitro folding experiments on circularized variants of YibK and YbeA, Mallam and co-workers discovered that the denatured ensembles, even in high concentrations of chemical denaturant under which conditions there was little or no secondary or tertiary structure, contained kinetically trapped knotted polypeptide chains [194]. It was then concluded that all the previous in vitro folding experiments on these recombinantly expressed and chemically denatured proteins actually probed refolding from an unfolded but knotted denatured state to a knotted and folded native structure. This unexpected result suggests that there are interactions in the denatured state that kinetically stabilize the knot. Although far-UV CD measurements indicate that there is no significant secondary structure present in the denatured state, recent backbone NMR assignments and chemical shifts of urea-denatured YbeA, show that, in fact, some residual secondary structure still remains under these conditions [203]. The fact that the knot can persist in the denatured state over a long period of time was also confirmed by another group who shared that equilibrium unfolding and refolding transitions of a structurally homologous MTase displayed apparent hysteresis [204]. This behaviour was speculated to be consistent with the uncoupling of the unfolding and untying events of the knotted protein [204]. Recently, single-molecule fluorescence resonance energy transfer (FRET) experiments were performed to characterise the denatured state of TrmD, another trefoil-knotted MTase [205]. Results suggested that the knot was not only retained under denaturing conditions (similar to that of YibK and YbeA) but also slid towards the C-terminus of the polypeptide chain during the unfolding process [205].

Up until recently, there have been no experimental studies into how the knot is first formed from an unknotted linear polypeptide chain. However, with the use of a coupled in vitro transcription-translation system and kinetic pulse-proteolysis experiments, Mallam and Jackson were able to specifically probe folding of nascent chains of YibK and YbeA after they were first synthesised by the ribosome (figure 11(a)) [199]. The results showed that the nascent chains could fold correctly to their trefoil-knotted structure, albeit very slowly. Moreover, a significant lag period between chain synthesis and emergence of a proteolytically stable native state was observed. The results were consistent with the protein knotting and folding from an initially unknotted nascent chain, thus demonstrating that a process associated with the knotting step is rate limiting. Additionally, the GroEL-GroES chaperonin was found to have a dramatic effect on the folding rate of the newly translated polypeptide chains, thus establishing that chaperonins are likely to be important in the post-translational folding of these bacterial knotted proteins in vivo.

Zoom In Zoom Out Reset image size

Very recently, we have investigated the knotting and folding behaviour of the nascent chains of the different N- and C-terminal This fusions of YibK and YbeA with the use of the coupled in vitro transcription-translation system and kinetic pulse-proteolysis experiments [206]. The results demonstrated that these multi-domain proteins with extremely deep knots can be synthesized in vitro and spontaneously knot without the help of any molecular chaperones, albeit very slowly. In addition, it was concluded that the C-terminus of these proteins is critical to the threading of the polypeptide chain to form the knot, thus providing the first experimental insight as to the mechanism of knotting for this class of bacterial knotted MTase. Further experiments with the GroEL-GroES chaperonin demonstrated that it actively assists the folding of knotted proteins by a mechanism that may involve the unfolding of kinetically trapped unknotted and misfolded intermediates (figure 11(b)). These key observations provide not only vital information into the complex folding pathway of trefoil-knotted proteins but also further insights into how topologically knotted proteins have withstood evolutionary pressures and achieve efficient folding in vivo.

In 2010, the Yeates group engineered an artificially trefoil-knotted protein by covalently linking together two monomers intertwined in the dimeric structure of HP0242 from H. pylori [207]. An in vitro experimental characterisation of this designed knotted protein and an unknotted monomeric variant of the HP0242 dimer was undertaken. Results showed that, although the knotted variant was more stable than the unknotted one, it folded at a considerably slower rate (approximately 20-fold), indicating that knotting, or some event associated with it, is likely rate-limiting.

AFM has also been used to study the mechanical unfolding of the shallow trefoil-knotted carbonic anhydrase B. In this case, the polypeptide chain was found to extend to a distance much shorter than its theoretical stretching length, indicating that the knotted structure is tightened but retained [182, 208]. Similarly, AFM mechanical unfolding experiments on the figure-of-eight knot in the chromophore-binding domain of the phytochrome also resulted in a tightened knot of approximately 17 residues [175]. Although these experiments do not necessarily provide extensive information on the folding pathways of these proteins, they were critical in demonstrating that the knots were present in the structure and in determining the minimum length of polypeptide chain required for knotting.

In addition to the trefoil-knotted proteins described in detail above, the other family of knotted proteins for which there has been any substantial experimental characterisation of their folding pathways are the 5₂-knotted UCHs [177, 209]. The unfolding of two human UCHs- UCH-L1, a neuronal form of the enzyme, and UCH-L3, ubiquitously expressed in many cell types, have been determined and, in both cases, the in vitro unfolding/refolding with chemical denaturants was shown to be fully reversible [177, 209]. In the case of UCH-L3, equilibrium unfolding data were fitted to a simple two-state model [209] whilst that for UCH-L1 were consistent with a three-state model in which an intermediate state is populated [177]. Using NMR hydrogen-deuterium exchange (HDX) experiments, the intermediate state was characterised indirectly and it was found that the central β-sheet core of the protein remains structured whilst many of the surrounding α-helices have unfolded [177]. Although a more complete analysis of the folding pathway of UCH-L1 has yet to be published, the folding is similar to UCH-L3, such that, both have multiple unfolding and refolding phases that indicate parallel pathways and the population of at least two, metastable intermediate states (Luo et al unpublished results).

5.3.2. Computational studies on knotted proteins.

Many computational studies have shed considerable light on the folding of knotted proteins. Coarse-grained simulations have been excellent at revealing the possible mechanism(s) and generic features of how knotted proteins fold [210, 211]. Wallin et al performed the first such simulation using a C_α model representation of YibK and, similar to experimental studies, observed two parallel folding pathways [210]. They also concluded that specific, non-native interactions involving residues in the C-terminal region of the chain were needed for the protein to knot and fold successfully. In contrast, Sulkowska and co-workers showed that native interactions alone are sufficient for simulating the folding of YibK and YbeA using a coarse-grained structure-based model, although the number of successful trajectories was only 1–2% [211]. These simulations also illustrated that partial unfolding (backtracking) events were needed because the order in which native contacts are formed is critical for the correct folding of the knotted structure and that folding frequently occurred through a slipknotted intermediate (figure 12(a)). Importantly, in the same study, simulations of a rewired, unknotted variant established that there are significant topological barriers in the folding of the knotted structure [211]. Using a similar model, initial results from recent kinetic unfolding simulations of a structurally homologous MTase revealed that unfolding of the protein to a fully unfolded, unknotted state occurs in a stepwise process [204]. In addition, the simulations showed that unknotting of the chain is slow compared to the initial unfolding [199].

Zoom In Zoom Out Reset image size

Similar computational approaches were also employed in the folding simulations of the 6₁-knot in DehI [154]. Although the probability of successful folds was low, the study revealed that the complex knotted structure can be formed by a simple tying process. In this case, two unknotted loops, a small loop and a larger loop (which includes a proline-rich unstructured region) are aligned and a knot can be formed by two alternative routes (figure 12(b)) [154]. In the first route, the C-terminus is threaded through the smaller loop (S-loop) via a slipknot conformation before the larger loop (B-loop) flips over the smaller loop. In the other route, the order of the two steps is reversed.

In contrast to very small proteins with simple architectures (which generally have fast unfolding and folding rates), all-atom molecular dynamics (MD) simulations have not been extensively applied to knotted systems, as they are frequently too large for such atomistic approaches to be used. However, it has been possible to use this method in a few cases on small, shallow knotted proteins, such as for MJ0366 from M. jannaschii, one of the smallest trefoil-knotted protein discovered to date [141]. Data from a thermodynamic analysis of the unfolding/folding revealed that the system is three-state, and an intermediate is first formed by twisting of a loop, followed by a rate-limiting step associated with the threading of the C-terminus through the loop. At temperatures near the folding temperature, two folding mechanisms were observed for the formation of the knotted native structure, whereby threading can occur via (i) a plugging route (the C-terminus goes through the knotting loop first) or (ii) the formation of a slipknot (figure 12(c)) [141]. Interestingly, lowering the temperature of the simulation resulted in mechanistic changes. These include a knotting via threading of the N-terminus and the 'backtracking' of misfolded proteins in topological traps. More recently, simulations on VirC2, a protein that has the same fold as MJ0366 but which possesses a deeper knot, also showed that it has a similar free energy profile, suggesting that topology plays a major role in the folding mechanism [212]. A Gō-like potential in which there is minimal energy frustration was also used to simulate the folding of a truncated mutant of another trefoil-knotted MTase [213]. Results from this study suggested a pathway in which the N-terminal region of the protein folds first and that threading of the C-terminus through the structure to form the knot is a late and rate-limiting step [213].

Molecular dynamics simulations were also used to simulate the high temperature unfolding of YibK [214]. The simulations revealed up to four intermediate states on the free energy landscape consistent with the parallel pathways and multiple intermediates observed in experimental studies. In addition, it was found that the denatured state of YibK only untied at very high simulation temperatures, when the C-terminus threads out of the knotting loop via a slipknot conformation. Other unfolding simulations have also been used to investigate the mechanical stability of knotted proteins and the effect of pulling position, pulling speed and temperature on the unfolding/untying of two other MTases [215]. It was shown that pulling the chain at both termini leads to the tightening of the knot whilst pulling at other positions can result in the unknotting of the chain (figure 12(d)).

Various computational studies have also employed Monte Carlo simulations on lattice models using Gō-like potentials to understand the folding mechanism of knotted proteins. In these cases, a potential based on a generic polymer model is used and additional attractive interactions are included for residues that are in contact with each other in the native state. Faisca and co-workers demonstrated that the folding of a model deeply knotted trefoil protein was much slower than a structurally similar but unknotted variant, and that knotting was a late event and concomitant with folding [216]. Using the same model, Soler and Faisca examined the effect of surface tethering on the folding of the system [217]. In this case, it was shown that the mobility of the terminus closest to the knot is critical for successful folding and hindrance results in a decrease in the folding rate and a change in the knotting pathway such that it involves threading of the other terminus. Recently, the same group extended these studies and used the same model to investigate in further detail the effect of knots, knot depth and motif on folding properties of 3₁-knotted proteins [180]. The results revealed that deeply knotted proteins have a higher probability of retaining their knots in the denatured ensemble, consistent with experimental studies. Furthermore, it was shown that specific native contacts within the trefoil-knotted core are crucial in maintaining the knot in the denatured state, and that threading occurs in the late stages of folding [180]. Most recently, Soler and co-workers extended their studies to investigate the folding mechanism of the more complex 5₂-knot [176]. Similar to the trefoil knots, it was shown that the chain terminus that is closest to the knotted core is important for the threading movement to form the knot and in no cases was a mechanism that involved the initial formation of a 3₁-knot observed. However, it was discovered that the probability of concomitant knotting and folding of 5₂-knotted proteins is significantly smaller than that for trefoil knots as threading to form the 5₂ knot is a particularly late conformational event [176].

Monte Carlo simulations of a C_α model of trefoil-knotted AOTCase showed that non-native contacts between the C-terminus and other regions in the protein are critical to form the knotting loop through which the chain is threaded [218], consistent with the study by Wallin and co-workers [210]. The importance of non-native interactions in promoting the folding of the native knotted topology of AOTCase and MJ036 was also recently highlighted in simulations employing protein models with different structural resolution (coarse-grained or atomistic) and various force fields (from pure native-centric to realistic atomistic ones) [219]. Again, it appears that these contacts were found to be between the C-terminus and a loop, through which the chain is threaded.

5.3.3. Experimental and computational studies on slipknots.

Despite the fact that there are now a considerable number of topologically knotted proteins in the PDB, it is worth noting that most proteins are unknotted. This suggests that evolution has, in general, avoided such structures. However, a recent study by Sulkowska and co-workers has established that, when they do occur, that both knotted and slipknotted topologies are conserved across different families despite very low sequence similarity [142]. Unsurprisingly, the parts of proteins which are strongly conserved are found within the knotted core and potential hinge regions which it has been speculated are important in the threading of the chain to form a knot or slipknot [142].

For some families of proteins, where there are a sizeable number of knotted and unknotted variants, it has been possible to undertake a phylogenetic analysis of the sequences, and thereby identify how knotted structures may have evolved from unknotted ancestors. Potestio and co-workers generated a phylogenetic tree of transcarbamylase-like folds [223]. In this case, it was known that some knotted and unknotted variants had different degrees of sequence identity suggesting pathways where structures and therefore sequences had diverged at different times. For example, the two knotted enzymes AOTCase and SOTCase share only 35% sequence identity [224] whilst the knotted AOTCase has 41% sequence identity with unknotted OTCase [225]. Reconstruction of the phylogenetic tree demonstrated that all the knotted homologues populate a sub-branch of the tree and that they differ from unknotted homologues by the presence of additional loop segments [223]. Thus, it has been suggested that some knotted structures have evolved from unknotted ones by the insertion of a 'knot-promoting' loop, which effectively encompasses another part of the chain thus forming the knot.

Loops have also been implicated in the formation of knotted structures from other studies. Virnau and co-workers used computational approaches to show that the knotted transcarbamylase AOTCase possesses a rather rigid proline-rich loop, which is lacking in the unknotted OTCase (figure 10(b)) [149]. Interestingly, the stevedore knot in α-haloacid dehalogenase DehI is also partly formed by a large proline-rich loop that links two unknotted regions within the structure [154].

Using a completely different approach, the group of Yeates have also demonstrated another route to knotted structures through the rational design of a novel knotted structure. In this case, a monomeric knotted protein was created by fusion of C- and N-terminal chains of a homodimer that forms a highly entangled but unknotted structure. This study demonstrated that the genetic fusion and tandem repeat of a gene of an unknotted dimeric protein could lead to trefoil-knotted structures [207].

It is clear that, once formed through some evolutionary pathway, knotted and slipknotted protein structures are highly conserved. However, through both experimental and computational studies, we also know that these types of structures have more complex folding pathways than their unknotted counterparts. This suggests that the knotted and slipknotted motifs within protein families may, in some way, be advantageous and important to either the function, or regulation, of the protein.

In summary, both experimental and computational studies have made significant progress in establishing some of the key general features of the folding pathways of topologically complex proteins. In contrast to small monomeric proteins with simple folds, it is clear that proteins with topologically knotted or slipknotted structures have much more complex energy landscapes with many intermediate states and parallel pathways. Computational studies have provided insights into the folding process, which may involve formation of a twisted loop followed by threading via an intermediate slipknot configuration, a plugging route or a 'flipping' mechanism, in which the knotting step may be rate-limiting [141, 211, 226]. In addition, it seems that non-native interactions may play a more important role for these types of structures with complex architectures than for the folding of smaller proteins with relatively simple folds [227–229]. Moreover, the formation of transient misfolded species that results in kinetic traps in the free energy landscape of topologically knotted proteins highly likely requires backtracking events and potentially the action of molecular chaperones so that the native structure can be both rapidly and efficiently achieved [199, 206, 211]. Such a 'frustrated' folding energy landscape is in contrast to the relatively smooth folding funnels proposed for smaller, simpler proteins [192, 230].

A number of recent studies have shown that knotted and slipknotted proteins are conserved suggesting that the knot, or slipknot, potentially play a role in the structure, stability, function or regulation of the protein. Despite this finding, it still has to be unambiguously established whether there are any advantageous properties of a knotted structure over an unknotted one. Indeed, whether there are any chemical or physical properties of such structures that are fundamentally different from unknotted ones. Understanding and identifying such properties will potentially provide key insights for future protein engineering applications and therapeutic developments.

The synthesis of molecular knots is challenging, as it requires defined pathways and (usually) entropically demanding transition states to achieve a specific knotted structure. Many early experimental efforts (albeit unsuccessful) and proposed synthetic routes towards molecular knots have provided significant insights into the problems of assembling such systems [250, 251]. Over the past two decades, the field of chemical topology has seen various synthetic strategies and approaches being employed for the preparation of different knotted molecules, many of which rely on template effects related to non-covalent interactions identified from supramolecular and coordination chemistry [16, 17, 248]. Here, we discuss these approaches and, in particular, compare the different mechanisms of knot formation using stepwise synthetic approaches to those of 'all-in one' strategies.

6.1.1. Metal template-based synthetic approaches.

Using an extension of Sauvage's original strategy for assembling [2] catenanes [239], Dietrich–Buchecker and Sauvage reported the first successful synthesis of a molecular trefoil knot in 1989 [244]. In this case, the end-groups of a dimetallic, double-stranded helicate, composed of two bisphenanthroline ligands and two copper(I) ions, were connected using Williamson ether synthesis. This generated the three crossing points needed for a trefoil knot; however, it was isolated in 3% yield only. A separate study by Dietrich–Buchecker and co-workers later showed that different spacers linking the phenanthroline groups were critical in determining the yield. In particular, the use of a rigid 1,3-phenylene spacer was found to assist in the stabilisation of the helicate assembly thus resulting in a yield of 29% [252]. However, it was not until the introduction of efficient catalysts for ring-closing olefin metathesis (RCM) that the best yield for a molecular trefoil knot (74%) was achieved (figure 14(a)) [253]. This successful approach was then extended to the preparation of composite knots, details of which can be found in [247]. In another case, the same group used octahedral iron(II) ions reacted with terpyridine-based ligands to template the synthesis of a trefoil knot [254]. The yield achieved, however, was significantly lower (20%), probably because the macrocyclisation was not as effective as that of the previous ligand-metal ion system. Through collaborative work, the groups of von Zelewsky and Sauvage were able to synthesise the first diastereospecific molecular trefoil knot in 74% yield by fusing chiral groups to a 2,2'-bipyridine ligand, thus controlling the stereochemistry of the two copper(I) ions to which the ligands were coordinated [255].

Zoom In Zoom Out Reset image size

In 2001, Hunter and co-workers reported the synthesis of a stable, 'open-knotted' structure, wherein a single linear tris-bipyridine ligand was coordinated around an octahedral zinc(II) ion [256]. This strategy directly relates to that published by Sokolov in 1973, when he first proposed that a trefoil knot motif could be achieved by arranging three bidentate ligands around an octahedral metal centre to generate the necessary crossings [257]. However, it was not until a decade later that the same group was able to produce the closed trefoil-knotted structure in 68% yield by trapping the acyclic complex through RCM and subsequent removal of the metal template (figure 14(b)) [258].

Active metal template strategies have also played a significant role in the preparation of interlocked compounds [238]. In this case, a metal ion acts simultaneously as a template as well as a catalyst for the synthesis of an entangled structure. In 2011, Leigh and co-workers used this strategy to synthesise the smallest molecular trefoil knot to date (a 76-atom long closed structure) in a yield of 24% [259]. A tetrahedral copper(I) ion acts as a template to coordinate a single polypyridyl ligand and form the crossing points, while another copper(I) ion binds to the functional end groups of the ligand, threads the loop through its coordination geometry and subsequently catalyses the covalent bond formation to create the trefoil knot motif (figure 14(c)).

Up until recently, the synthesis of molecular knots via a metal-based template strategy has been mainly performed with transition metals. However, recently, with the use of a lanthanide (Ln³⁺) ion, Leigh and co-workers demonstrated that it can template three 2,6-pyridinedicarboxamide ligands to which subsequent cyclisation by RCM resulted in an 81-atom loop trefoil knot molecule isolated in 58% yield (figure 14(d)) [260].

In these metal-based template approaches, the molecular knots are clearly formed in a stepwise manner, whereby the ligand(s) are first coordinated to metal ion(s). In some cases, this step results in a molecule in which the single ligand assembles around a central metal ion in such a way that there are a number of crossings of regions of the ligand. Alternatively, a number of ligands preassemble around the central ion(s) resulting in crossings of the individual building blocks. In other cases, there is a threading event through a loop created by the initial metal-ligand complex. In all cases, covalent linkage of either the termini of a single ligand or the monomeric units results in a closed knotted structure.

6.1.2. Hydrogen-bond template approaches.

Although not as frequently used as the metal-based template strategies, amide–amide hydrogen bonding interactions have also been shown to be important in the synthesis of molecular knots. In 1994, Hunter and co-workers used this approach to produce what they thought was a [2] catenane from the reaction of a diamine and a diacyl chloride [261]. However, several years later, Vögtle and co-workers repeated this one-step synthesis and with the use of x-ray crystallography, discovered that the resultant molecule was, in fact, a trefoil knot [262]. It was then suggested that it was highly likely that the linear diamine, composed of three units of the diamine and two units of diacyl, forms first, then folds into a helical loop which subsequently self-threads its remaining part through the loop. A reaction between the remaining carboxylic acid chloride unit and the terminal amino groups of the open loop then results in the closing of the loop to form the trefoil knot in 20% yield (figure 15(a)) [242, 262]. This synthetic approach highlights the importance of intra-molecular hydrogen bonding in the loop for subsequent knot formation.

Zoom In Zoom Out Reset image size

In 2006, Feigel and co-workers reported the synthesis of a molecular trefoil knot in 21% yield, which also made use of amide-amide hydrogen bonding interactions [263]. In this case, the trefoil knot was formed unexpectedly during the amide coupling reaction of 3-α-aminodeoxycholanic acid with L-valine. Similar to the previous synthesis, this is a one-pot procedure in which no external templating agent was needed to form the knotted architecture.

6.1.3. Dynamic combinatorial library (DCL) approaches.

6.1.4. Other synthetic approaches to molecular knots.

Chirality is ubiquitous in chemistry, and knots are often chiral species. If the pure topological enantiomers of such can be obtained from the resolution of racemates, they will have specific optical properties. In many cases, it has been possible to isolate enantiomerically pure species. For example, enantiomers can be separated with the use of chiral HPLC [270, 271]. In another case, Sauvage and von Zelewsky were able to specifically form a single enantiomer by controlling the stereochemistry of the chiral helicate precursor [255]. Sanders and co-workers were also able to stereoselectively synthesise a trefoil knot by constraining the chirality of the building block in the DCL approach [6]. Recently, Leigh and co-workers who reported the synthesis of a lanthanide-templated molecular trefoil knot speculated that its chirality may influence the photophysical properties of the encapsulated lanthanide ion [260].

The study of the conformational properties of intertwined molecules is also of great interest due to their potential applications in the assembly of molecular switches. As molecular knots are increasingly becoming targets of chemical synthesis, it is important to understand what kind of motion is expected from the knotted topology. A study by Sauvage's group compared the dynamics of two different types of molecular trefoil knots formed by the metal-template based approach, in which the phenanthroline units were linked either by oligomethylene or m-phenylene spacers. In both cases, the molecular knots which still contained copper(I) ions were found to be generally rigid in solution [272]. However, removal of the metal ions led to rearrangement of the knotted backbone and, in both cases resulted in different dynamic behaviour. Those molecules containing the oligomethylene linkers had significantly greater conformational mobility in solution in comparison to those with m-phenylene spacers [244]. This study also showed that the conformational rigidity of partially or fully demetalated molecular knots can be restored again after re-complexation [272].

How do the conformational dynamics of the amide molecular knots formed via hydrogen bond interactions compare to those of the phenanthroline molecular knots? Based on ¹H- and ³¹P-NMR spectroscopic measurements, Vögtle and co-workers reported that the amide molecular trefoil knots retain relatively rigid, non-symmetrical structures in DMSO, even though no metal ion is present [242]. However, addition of other solvents to the solution of these knots rapidly resulted in conformational change, and, in some cases, led to increased flexibility or increased rigidity [242]. Such changes in dynamics brought about by change in solution conditions makes these systems interesting for the development of molecular switches. In another case, the organic trefoil knot synthesised using the DCL approach exhibited sharp NMR signals in water demonstrating that the molecule was relatively rigid under these conditions. The signals remained unchanged upon increasing the temperature (from 298 to 358 K) or adding acetonitrile (from 0 to 50%), indicating that the structure is sufficiently stable such that it does not undergo gross conformational change upon a change in conditions [6], in contrast to the trefoil knots synthesised and studied by the Sauvage and Vögtle groups. Such conformational rigidity was also observed in the highly symmetric, achiral figure-of-eight knot synthesised using the DCL approach [264].

Although challenging, recently, chemists have successfully developed a number of different experimental strategies for the creation of molecular knots. These approaches have been used to synthesise a number of linked species, including Solomon links, Borromean rings, and a Star of David catenane. However, they have also been employed to make true knotted molecules including a 3₁, 4₁ and 5₁ knot. The different synthetic strategies can generally be considered as either a template-based method (for example the metal-based templates), or those which use hydrogen bonding or π  −  π interactions to first preassemble the building block(s) in such a way that covalent linkage of either the termini or of the monomer units results in a knotted structure. Alternatively, DCL-based approaches utilise the fact that a number of building blocks can come together to form chains of different lengths which can then fold to a thermodynamically stable state. In the first case, there need not be any threading event, but preassembly is crucial, whilst in the DCL approach, threading can occur. The properties of the molecular knots created using synthetic strategies are beginning to become established. Whereas, in some cases, the molecules are rigid in the presence of the templating metal ion, they can clearly undergo conformational change and their flexibility can alter when the metal ion is removed. The dynamics of such systems have also been found to vary depending upon environmental conditions. In other cases, such as those knotted molecules created by the DCL approach, which favours thermodynamically stable states, the evidence suggests these are relatively rigid molecules whose structures do not change significantly with environmental conditions.

As more and more new topologically complex structures are created, this raises the issue of whether knotting, linking, etc convey novel or important properties on the molecule. If they do, then it may be possible to exploit them in practical applications such as materials and pharmaceuticals. Readers who are interested in comprehensive discussion of these synthetic approaches are directed to the following reviews [16, 17, 241, 242, 248].