Molecular knots in biology and chemistry

Molecular entanglement can strongly increase basicity

Brønsted basicity is a fundamental chemical property featured by several kinds of inorganic and organic compounds. In this Review, we treat a particularly high basicity resulting from the mechanical entanglement involving two or more molecular subunits in catenanes and rotaxanes. Such entanglement allows a number of basic sites to be in close proximity with each other, highly increasing the proton affinity in comparison with the corresponding, non-entangled counterparts up to obtain superbases, properly defined as mechanically interlocked superbases. In the following pages, the development of this kind of superbases will be described with a historical perusal, starting from the initial, serendipitous findings up to the most recent reports where the strong basic property of entangled molecular units is the object of a rational design.

Mechanical scission of a knotted polymer

Molecular knots and entanglements form randomly and spontaneously in both biological and synthetic polymer chains. It is known that macroscopic materials, such as ropes, are substantially weakened by the presence of knots, but until now it has been unclear whether similar behaviour occurs on a molecular level. Here we show that the presence of a well-defined overhand knot in a polymer chain substantially increases the rate of scission of the polymer under tension (≥2.6× faster) in solution, because deformation of the polymer backbone induced by the tightening knot activates otherwise unreactive covalent bonds. The fragments formed upon severing of the knotted chain differ from those that arise from cleavage of a similar, but unknotted, polymer. Our solution studies provide experimental evidence that knotting can contribute to higher mechanical scission rates of polymers. It also demonstrates that entanglement design can be used to generate mechanophores that are among the most reactive described to date, providing opportunities to increase the reactivity of otherwise inert functional groups.

Tying a true topological protein knot by cyclization

Knotted proteins are fascinating to biophysicists because of their robust ability to fold into intricately defined three-dimensional structures with complex and topologically knotted arrangements. Exploring the biophysical properties of the knotted proteins is of significant interest, as they could offer enhanced chemical, thermal, and mechanostabilities. A true mathematical knot requires a closed path; in contrast, knotted protein structures have open N- and C-termini. To address the question of how a truly knotted protein differs from the naturally occurring counterpart, we enzymatically cyclized a 31 knotted YibK protein from Haemophilus influenza (HiYibK) to investigate the impact of path closure on its structure-function relationship and folding stability. Through the use of a multitude of structural and biophysical tools, including X-ray crystallography, NMR spectroscopy, small angle X-ray scattering, differential scanning calorimetry, and isothermal calorimetry, we showed that the path closure minimally perturbs the native structure and ligand binding of HiYibK. Nevertheless, the cyclization did alter the folding stability and mechanism according to chemical and thermal unfolding analysis. These molecular insights contribute to our fundamental understanding of protein folding and knotting that could have implications in the protein design with higher stabilities.

Iminopyrrole-Based Self-Assembly: A Route to Intrinsically Flexible Molecular Links and Knots

The use of 2,5-diformylpyrrole in self-assembly reactions with diamines and Zn(II)/Cd(II) salts allowed the preparation of [2]catenane, trefoil knot, and Borromean rings. The intrinsically dynamic nature of the diiminopyrrole motif rendered all of the formed assemblies intramolecularly flexible. The presence of diiminopyrrole revealed new coordination motifs and influenced the host-guest chemistry of the systems, as illustrated by hexafluorophosphate encapsulation by Borromean rings.

Self-Assembly of an [M8 L24 ]16+ Intertwined Cube and a Giant [M12 L16 ]24+ Orthobicupola

Through coordination-driven self-assembly, aesthetically captivating structures can be formed by tuning the length or flexibility of various components. The self-assembly of an elongated rigid terphenyl-based tetra-pyridyl ligand (L1) with a cis-Pd(II) acceptor produces an [M12 L16 ]24+ triangular orthobicupola structure (1). When flexibility is introduced into the ligand by the incorporation of a -CH2 - group between the dipyridylamine and terphenyl rings in the ligand (L2), anunique [M8 L24 ]16+ water-soluble 'intertwined cubic structure' (2) results. The inherent flexibility of ligand L2 might be the key factor behind the formation of the thermodynamically stable and 'intertwined cubic structure' in this scenario. This research showcases the ability to design and fabricate novel, topologically distinctive molecular structures by a straightforward and efficient approach.

Structure, dynamics, and stability of the smallest and most complex 71 protein knot

Proteins can spontaneously tie a variety of intricate topological knots through twisting and threading of the polypeptide chains. Recently developed artificial intelligence algorithms have predicted several new classes of topological knotted proteins, but the predictions remain to be authenticated experimentally. Here, we showed by X-ray crystallography and solution-state NMR spectroscopy that Q9PR55, an 89-residue protein from Ureaplasma urealyticum, possesses a novel 71 knotted topology that is accurately predicted by AlphaFold 2, except for the flexible N terminus. Q9PR55 is monomeric in solution, making it the smallest and most complex knotted protein known to date. In addition to its exceptional chemical stability against urea-induced unfolding, Q9PR55 is remarkably robust to resist the mechanical unfolding-coupled proteolysis by a bacterial proteasome, ClpXP. Our results suggest that the mechanical resistance against pulling-induced unfolding is determined by the complexity of the knotted topology rather than the size of the molecule.

Resolution of Expanded Porphyrinoids: A Path to Persistent Chirality and Appealing Chiroptical Properties

Chirality is a fundamental characteristic of nature. Expanded porphyrinoids and their analogues offer an attractive platform for delving into the intricacies of chirality. Expanded porphyrinoids comprise pyrrolic macrocycles and related heterocyclic systems. As a class, expanded porphyrinoids are widely recognized for their flexible structural features, nontrivial coordination capabilities, and intriguing optical and electronic properties. With limited exceptions, their inherent conformational flexibility coupled with a low racemization barrier allows for the facile interchange between enantiomers. As a result, achieving the effective chiral resolution of individual enantiomers and the subsequent exploration of their chiroptical properties represents a significant challenge. This review summarizes strategies used to realize the chiral resolution of expanded porphyrinoids and the understanding of intrinsic chiroptical properties that has emerged from these separation efforts. It is our hope that this review will serve not only to codify our current understanding of chiral expanded porphyrinoids, but also inspire advances in the generalized area of chiral functional materials.

Folding and functions of knotted proteins

Topologically knotted proteins have entangled structural elements within their native structures that cannot be disentangled simply by pulling from the N- and C-termini. Systematic surveys have identified different types of knotted protein structures, constituting as much as 1% of the total entries within the Protein Data Bank. Many knotted proteins rely on their knotted structural elements to carry out evolutionarily conserved biological functions. Being knotted may also provide mechanical stability to withstand unfolding-coupled proteolysis. Reconfiguring a knotted protein topology by circular permutation or cyclization provides insights into the importance of being knotted in the context of folding and functions. With the explosion of predicted protein structures by artificial intelligence, we are now entering a new era of exploring the entangled protein universe.

AlphaFold Blindness to Topological Barriers Affects Its Ability to Correctly Predict Proteins' Topology

AlphaFold is a groundbreaking deep learning tool for protein structure prediction. It achieved remarkable accuracy in modeling many 3D structures while taking as the user input only the known amino acid sequence of proteins in question. Intriguingly though, in the early steps of each individual structure prediction procedure, AlphaFold does not respect topological barriers that, in real proteins, result from the reciprocal impermeability of polypeptide chains. This study aims to investigate how this failure to respect topological barriers affects AlphaFold predictions with respect to the topology of protein chains. We focus on such classes of proteins that, during their natural folding, reproducibly form the same knot type on their linear polypeptide chain, as revealed by their crystallographic analysis. We use partially artificial test constructs in which the mutual non-permeability of polypeptide chains should not permit the formation of complex composite knots during natural protein folding. We find that despite the formal impossibility that the protein folding process could produce such knots, AlphaFold predicts these proteins to form complex composite knots. Our study underscores the necessity for cautious interpretation and further validation of topological features in protein structures predicted by AlphaFold.

Knot Formation on DNA Pushed Inside Chiral Nanochannels

We performed coarse-grained molecular dynamics simulations of DNA polymers pushed inside infinite open chiral and achiral channels. We investigated the behavior of the polymer metrics in terms of span, monomer distributions and changes of topological state of the polymer in the channels. We also compared the regime of pushing a polymer inside the infinite channel to the case of polymer compression in finite channels of knot factories investigated in earlier works. We observed that the compression in the open channels affects the polymer metrics to different extents in chiral and achiral channels. We also observed that the chiral channels give rise to the formation of equichiral knots with the same handedness as the handedness of the chiral channels.

Di-2,7-pyrenidecaphyrin(1.1.0.0.0.1.1.0.0.0) and Its Bis-Organopalladium Complexes: Synthesis and Chiroptical Properties

A non-aromatic expanded carbaporphyrinoid, incorporating two built-in 2,7-pyrenylene moieties was synthesized. The intrinsically labile structure was demonstrated by proton-triggered conformational changes between the figure-of-eight and quasi-Möbius conformers. Upon treatment with Pd(OAc)₂ , the reaction produces two bis-Pd^II complexes with distinct coordination modes. Metal coordination serves to fix the macrocyclic frameworks with the net result that both bis-Pd^II complexes could be resolved by high performance liquid chromatography (HPLC) on a chiral stationary phase. The isolated enantiomers showed persistent chiroptical properties as evidenced by the intense response in the circular dichroism (CD) spectra and the record high absorption dissymmetry factors (g_abs of up to 0.038) seen in the near-infrared spectral region. Moreover, the mutual interconversion of these two Pd^II complexes was found to be stereospecific and to favor the more stable isomers under weakly acidic conditions.

Computational Design of Extraordinarily Stable Peptide Structures through Side-Chain-Locked Knots

Extraordinarily stable protein and peptide structures are critically demanded in many applications. Typical approaches to enhance protein and peptide stability are strengthening certain interactions. Here, we develop a very different approach: stabilizing peptide structures through side-chain-locked knots. More specifically, a peptide core consists of a knot, which is prevented from unknotting and unfolding by large side chains of amino acids at knot boundaries. These side chains impose free energy barriers for unknotting. The free energy barriers are quantified using all-atom and coarse-grained simulations. The barriers become infinitely high for large side chains and tight knot cores, resulting in stable peptide structures, which never unfold unless one chemical bond is broken. The extraordinary stability is essentially kinetic stability. Our new approach lifts the thermodynamic restriction in designing peptide structures, provides extra freedom in selecting sequence and structural motifs that are thermodynamically unstable, and should expand the functionality of peptides. This work also provides a bottom-up understanding of how knotting enhances protein stability.

AlphaFold predicts the most complex protein knot and composite protein knots

The computer artificial intelligence system AlphaFold has recently predicted previously unknown three‐dimensional structures of thousands of proteins. Focusing on the subset with high‐confidence scores, we algorithmically analyze these predictions for cases where the protein backbone exhibits rare topological complexity, that is, knotting. Amongst others, we discovered a 7₁‐knot, the most topologically complex knot ever found in a protein, as well several six‐crossing composite knots comprised of two methyltransferase or carbonic anhydrase domains, each containing a simple trefoil knot. These deeply embedded composite knots occur evidently by gene duplication and interconnection of knotted dimers. Finally, we report two new five‐crossing knots including the first 5₁‐knot. Our list of analyzed structures forms the basis for future experimental studies to confirm these novel‐knotted topologies and to explore their complex folding mechanisms.

The dynamic, motile and deformative properties of RNA nanoparticles facilitate the third milestone of drug development

Besides mRNA, rRNA, and tRNA, cells contain many other noncoding RNA that display critical roles in the regulation of cellular functions. Human genome sequencing revealed that the majority of non-protein-coding DNA actually codes for non-coding RNAs. The dynamic nature of RNA results in its motile and deformative behavior. These conformational transitions such as the change of base-pairing, breathing within complemented strands, and pseudoknot formation at the 2D level as well as the induced-fit and conformational capture at the 3D level are important for their biological functions including regulation, translation, and catalysis. The dynamic, motile and catalytic activity has led to a belief that RNA is the origin of life. We have recently reported that the deformative property of RNA nanoparticles enhances their penetration through the leaky blood vessel of cancers which leads to highly efficient tumor accumulation. This special deformative property also enables RNA nanoparticles to pass the glomerulus, overcoming the filtration size limit, resulting in fast renal excretion and rapid body clearance, thus low or no toxicity. The biodistribution of RNA nanoparticles can be further improved by the incorporation of ligands for cancer targeting. In addition to the favorable biodistribution profiles, RNA nanoparticles possess other properties including self-assembly, negative charge, programmability, and multivalency; making it a great material for pharmaceutical applications. The intrinsic negative charge of RNA nanoparticles decreases the toxicity of drugs by preventing nonspecific binding to the negative charged cell membrane and enhancing the solubility of hydrophobic drugs. The polyvalent property of RNA nanoparticles allows the multi-functionalization which can apply to overcome drug resistance. This review focuses on the summary of these unique properties of RNA nanoparticles, which describes the mechanism of RNA dynamic, motile and deformative properties, and elucidates and prepares to welcome the RNA therapeutics as the third milestone in pharmaceutical drug development.

Methods to study folding of alpha-helical membrane proteins in lipids

How alpha-helical membrane proteins fold correctly in the highly hydrophobic membrane interior is not well understood. Their folding is known to be highly influenced by the lipids within the surrounding bilayer, but the majority of folding studies have focused on detergent-solubilized protein rather than protein in a lipid environment. There are different ways to study folding in lipid bilayers, and each method has its own advantages and disadvantages. This review will discuss folding methods which can be used to study alpha-helical membrane proteins in bicelles, liposomes, nanodiscs or native membranes. These folding methods include in vitro folding methods in liposomes such as denaturant unfolding studies, and single-molecule force spectroscopy studies in bicelles, liposomes and native membranes. This review will also discuss recent advances in co-translational folding studies, which use cell-free expression with liposomes or nanodiscs or are performed in vivo with native membranes.

A trefoil knot self-templated through imination in water

The preparation of topologically nontrivial molecules is often assisted by covalent, supramolecular or coordinative templates that provide spatial pre-organization for all components. Herein, we report a trefoil knot that can be self-assembled efficiently in water without involving additional templates. The direct condensation of three equivalents of a tetraformyl precursor and six equivalents of a chiral diamine produces successfully a [3 + 6] trefoil knot whose intrinsic handedness is dictated by the stereochemical configuration of the diamine linkers. Contrary to the conventional wisdom that imine condensation is not amenable to use in water, the multivalent cooperativity between all the imine bonds within the framework makes this trefoil knot robust in the aqueous environment. Furthermore, the presence of water is proven to be essential for the trefoil knot formation. A topologically trivial macrocycle composed of two tetraformyl and four diamino building blocks is obtained when a similar reaction is performed in organic media, indicating that hydrophobic effect is a major driving force behind the scene.

Self-Assembly of Molecular Trefoil Knots Featuring Pentadecanuclear Homoleptic AuI -, AuI /AgI -, or AuI /CuI -Alkynyl Coordination

Metal-free and metal-containing molecular trefoil knots are fascinating ensembles that are usually covalently assembled, the latter requiring the rational design of di- or multidentate/multipodal ligands as connectors. In this work, we describe the self-assembly of pentadecanuclear Au^I trefoil knots [Au₁₅ (C≡CR)₁₅ ] from monoalkynes HC≡CR (R=9,9-X₂ -fluorenyl with X=nBu, n-hexyl) and [Au^I (THT)Cl]. Hetero-bimetallic counterparts [Au₉ M₆ (C≡CR)₁₅ ] (M=Cu/Ag) were self-assembled by reactions of [Au₁₅ (C≡CR)₁₅ ] with [Cu(MeCN)₄ ]⁺ /AgNO₃ and HC≡CR. The type of pentadecanuclear trefoil knots described herein is characterized by X-ray crystallography, 2D NMR and HR-ESI-MS. [Au₉ Cu₆ (C≡CR)₁₅ ] is relatively stable in hexane; its excited state properties were investigated. DFT calculations revealed that non-covalent metal-metal and metal-ligand interactions, together with longer alkyl chain-strengthened inter-ligand dispersion interactions, govern the stability of the trefoil knot structures.

Investigation of the structural dynamics of a knotted protein and its unknotted analog using molecular dynamics

The role of knots in proteins remains elusive. Some studies suggest an impact on stability; the difficulty in comparing systems to assess this effect, however, has been a significant challenge. In this study, we produced and analyzed molecular dynamic trajectories considering three different temperatures of two variants of ornithine transcarbamylase (OTC), only one of which has a 3₁ knot, in order to evaluate the relative stability of the two molecules. RMSD showed equilibrated structures for the produced trajectories, and RMSF showed subtle differences in flexibility. In the knot moiety, the knotted protein did not show a great deal of fluctuation at any temperature. For the unknotted protein, the residue GLY243 showed a high fluctuation in the corresponding moiety. The fraction of native contacts (Q) showed a similar profile at all temperatures, with the greatest decrease by 436 K. The investigation of conformational behavior with principal component analysis (PCA) and dynamic cross-correlation map (DCCM) showed that knotted protein is less likely to undergo changes in its conformation under the conditions employed compared to unknotted. PCA data showed that the unknotted protein had greater dispersion in its conformations, which suggests that it has a greater capacity for conformation transitions in response to thermal changes. DCCM graphs comparing the 310 K and 436 K temperatures showed that the knotted protein had less change in its correlation and anti-correlation movements, indicating stability compared to the unknotted.

Vernier template synthesis of molecular knots

Molecular knots are often prepared using metal helicates to cross the strands. We found that coordinatively mismatching oligodentate ligands and metal ions provides a more effective way to synthesize larger knots using Vernier templating. Strands composed of different numbers of tridentate 2,6-pyridinedicarboxamide groups fold around nine-coordinate lanthanide (III) ions to generate strand-entangled complexes with the lowest common multiple of coordination sites for the ligand strands and metal ions. Ring-closing olefin metathesis then completes the knots. A 3:2 (ditopic strand:metal) Vernier assembly produces +3₁#+3₁ and -3₁#-3₁ granny knots. Vernier complexes of 3:4 (tetratopic strand:metal) stoichiometry selectively form a 378-atom-long trefoil-of-trefoils triskelion knot with 12 alternating strand crossings or, by using opposing stereochemistry at the terminus of the strand, an inverted-core triskelion knot with six alternating and six nonalternating strand crossings.

Cell-Free Expression to Probe Co-Translational Insertion of an Alpha Helical Membrane Protein

The majority of alpha helical membrane proteins fold co-translationally during their synthesis on the ribosome. In contrast, most mechanistic folding studies address refolding of full-length proteins from artificially induced denatured states that are far removed from the natural co-translational process. Cell-free translation of membrane proteins is emerging as a useful tool to address folding during translation by a ribosome. We summarise the benefits of this approach and show how it can be successfully extended to a membrane protein with a complex topology. The bacterial leucine transporter, LeuT can be synthesised and inserted into lipid membranes using a variety of in vitro transcription translation systems. Unlike major facilitator superfamily transporters, where changes in lipids can optimise the amount of correctly inserted protein, LeuT insertion yields are much less dependent on the lipid composition. The presence of a bacterial translocon either in native membrane extracts or in reconstituted membranes also has little influence on the yield of LeuT incorporated into the lipid membrane, except at high reconstitution concentrations. LeuT is considered a paradigm for neurotransmitter transporters and possesses a knotted structure that is characteristic of this transporter family. This work provides a method in which to probe the formation of a protein as the polypeptide chain is being synthesised on a ribosome and inserting into lipids. We show that in comparison with the simpler major facilitator transporter structures, LeuT inserts less efficiently into membranes when synthesised cell-free, suggesting that more of the protein aggregates, likely as a result of the challenging formation of the knotted topology in the membrane.

Quantifying the effects of slit confinement on polymer knots using the tube model

Knots can spontaneously form in DNA, proteins, and other polymers and affect their properties. These knots often experience spatial confinement in biological systems and experiments. While confinement dramatically affects the knot behavior, the physical mechanisms underlying the confinement effects are not fully understood. In this work, we provide a simple physical picture of the polymer knots in slit confinement using the tube model. In the tube model, the polymer segments in the knot core are assumed to be confined in a virtual tube due to the topological restriction. We first perform Monte Carlo simulation of a flexible knotted chain confined in a slit. We find that with the decrease of the slit height from H=+∞ (the 3D case) to H=2a (the 2D case), the most probable knot size L_{knot}^{*} dramatically shrinks from (L_{knot}^{*})_{3D}≈140a to (L_{knot}^{*})_{2D}≈26a, where a is the monomer diameter of the flexible chain. Then we quantitatively explain the confinement-induced knot shrinking and knot deformation using the tube model. Our results for H=2a can be applied to a polymer knot on a surface, which resembles DNA knots measured by atomic force microscopy under the conditions that DNA molecules are weakly absorbed on the surface and reach equilibrium 2D conformations. This work demonstrates the effectiveness of the tube model in understanding polymer knots.

Circuit Topology for Bottom-Up Engineering of Molecular Knots

The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular functions, and their entanglement is believed to provide them with thermal and kinetic stability. Yet, little is known about the design principles of naturally evolved molecular knots. Intra-chain contacts and chain entanglement contribute to the folding of knotted proteins. Circuit topology, a theory that describes intra-chain contacts, was recently generalized to account for chain entanglement. This generalization is unique to circuit topology and not motivated by other theories. In this conceptual paper, we systematically analyze the circuit topology approach to a description of linear chain entanglement. We utilize a bottom-up approach, i.e., we express entanglement by a set of four fundamental structural units subjected to three (or five) binary topological operations. All knots found in proteins form a well-defined, distinct group which naturally appears if expressed in terms of these basic structural units. We believe that such a detailed, bottom-up understanding of the structure of molecular knots should be beneficial for molecular engineering.

Channels with Helical Modulation Display Stereospecific Sensitivity for Chiral Superstructures

By means of coarse-grained molecular dynamics simulations, we explore chiral sensitivity of confining spaces modelled as helical channels to chiral superstructures represented by polymer knots. The simulations show that helical channels exhibit stereosensitivity to chiral knots localized on linear chains by effect of external pulling force and also to knots embedded on circular chains. The magnitude of the stereoselective effect is stronger for torus knots, the effect is weaker in the case of twist knots, and amphichiral knots do exhibit no chiral effects. The magnitude of the effect can be tuned by the so-far investigated radius of the helix, the pitch of the helix and the strength of the pulling force. The model is aimed to simulate and address a range of practical situations that may occur in experimental settings such as designing of nanotechnological devices for the detection of topological state of molecules, preparation of new gels with tailor made stereoselective properties, or diffusion of knotted DNA in biological conditions.

Entanglements of structure elements revealed in RNA 3D models

Computational methods to predict RNA 3D structure have more and more practical applications in molecular biology and medicine. Therefore, it is crucial to intensify efforts to improve the accuracy and quality of predicted three-dimensional structures. A significant role in this is played by the RNA-Puzzles initiative that collects, evaluates, and shares RNAs built computationally within currently nearly 30 challenges. RNA-Puzzles datasets, subjected to multi-criteria analysis, allow revealing the strengths and weaknesses of computer prediction methods. Here, we study the issue of entangled RNA fragments in the predicted RNA 3D structure models. By entanglement, we mean an arrangement of two structural elements such that one of them passes through the other. We propose the classification of entanglements driven by their topology and components. It distinguishes two general classes, interlaces and lassos, and subclasses characterized by element types-loops, dinucleotide steps, open single-stranded fragments-and puncture multiplicity. Our computational pipeline for entanglement detection, applied for 1,017 non-redundant models from RNA-Puzzles, has shown the frequency of different entanglements and allowed identifying 138 structures with intersected assemblies.

Tying up the Loose Ends: A Mathematically Knotted Protein

Knots have attracted scientists in mathematics, physics, biology, and engineering. Long flexible thin strings easily knot and tangle as experienced in our daily life. Similarly, long polymer chains inevitably tend to get trapped into knots. Little is known about their formation or function in proteins despite >1,000 knotted proteins identified in nature. However, these protein knots are not mathematical knots with their backbone polypeptide chains because of their open termini, and the presence of a “knot” depends on the algorithm used to create path closure. Furthermore, it is generally not possible to control the topology of the unfolded states of proteins, therefore making it challenging to characterize functional and physicochemical properties of knotting in any polymer. Covalently linking the amino and carboxyl termini of the deeply trefoil-knotted YibK from Pseudomonas aeruginosa allowed us to create the truly backbone knotted protein by enzymatic peptide ligation. Moreover, we produced and investigated backbone cyclized YibK without any knotted structure. Thus, we could directly probe the effect of the backbone knot and the decrease in conformational entropy on protein folding. The backbone cyclization did not perturb the native structure and its cofactor binding affinity, but it substantially increased the thermal stability and reduced the aggregation propensity. The enhanced stability of a backbone knotted YibK could be mainly originated from an increased ruggedness of its free energy landscape and the destabilization of the denatured state by backbone cyclization with little contribution from a knot structure. Despite the heterogeneity in the side-chain compositions, the chemically unfolded cyclized YibK exhibited several macroscopic physico-chemical attributes that agree with theoretical predictions derived from polymer physics.

Converging experimental and computational views of the knotting mechanism of a small knotted protein

MJ0366 from Methanocaldococcus jannaschii is the smallest topologically knotted protein known to date. 92 residues in length, MJ0366 ties a trefoil (31) knot by threading its C-terminal helix through a buttonhole formed by the remainder of the secondary structure elements. By generating a library of point mutations at positions pertinent to the knot formation, we systematically evaluated the contributions of individual residues to the folding stability and kinetics of MJ0366. The experimental Φ-values were used as restraints to computationally generate an ensemble of conformations that correspond to the transition state of MJ0366, which revealed several nonnative contacts. The importance of these nonnative contacts in stabilizing the transition state of MJ0366 was confirmed by a second round of mutagenesis, which also established the pivotal role of F15 in stapling the network of hydrophobic interactions around the threading C-terminal helix. Our converging experimental and computational results show that, despite the small size, the transition state of MJ0366 is formed at a very late stage of the folding reaction coordinate, following a polarized pathway. Eventually, the formation of extensive native contacts, as well as a number of nonnative ones, leads to the threading of the C-terminal helix that defines the topological knot.

Time-dependent knotting of agitated chains

Agitated strings serve as macroscale models of spontaneous knotting, providing valuable insight into knotting dynamics at the microscale while allowing explicit analysis of the resulting knot topologies. We present an experimental setup for confined macroscale knot formation via tumbling along with a software interface to process complex knot data. Our setup allows characterization of knotting probability, knot complexity, and knot formation dynamics for knots with as many as 50 crossings. We find that the probability of knotting saturates below 80% within 100 s of the initiation of tumbling and that this saturation probability does not increase for chains above a critical length, an indication of nonequilibrium knot-formation conditions in our experiment. Despite the saturation in knot formation, we show that longer chains, while being more confined, will always tend to form knots of higher complexity since the free end can access a greater number of loops during tumbling.

Spontaneous knotting of a flexible fiber in chaotic flows

We consider the problem of an inextensible but flexible fiber advected by a steady chaotic flow, and ask the simple question of whether the fiber can spontaneously knot itself. Using a one-dimensional Cosserat model, a simple local viscous drag model and discrete contact forces, we explore the probability of finding knots at any given time when the fiber is interacting with the ABC class of flows. The bending rigidity is shown to have a marginal effect compared to that of increasing the fiber length. Complex knots are formed up to 11 crossings, but some knots are more probable than others. The finite-time Lyapunov exponent of the flow is shown to have a positive effect on the knot probability. Finally, contact forces appear to be crucial since knotted configurations can remain stable for times much longer than the turnover time of the flow, something that is not observed when the fiber can freely cross itself.

A molecular endless (74) knot

Current strategies for the synthesis of molecular knots focus on twisting, folding and/or threading molecular building blocks. Here we report that Zn(II) or Fe(II) ions can be used to weave ligand strands to form a woven 3 × 3 molecular grid. We found that the process requires tetrafluoroborate anions to template the assembly of the interwoven grid by binding within the square cavities formed between the metal-coordinated criss-crossed ligands. The strand ends of the grid can subsequently be joined through within-grid alkene metathesis reactions to form a topologically trivial macrocycle (unknot), a doubly interlocked [2]catenane (Solomon link) and a knot with seven crossings in a 258-atom-long closed loop. This 7₄ knot topology corresponds to that of an endless knot, which is a basic motif of Celtic interlace, the smallest Chinese knot and one of the eight auspicious symbols of Buddhism and Hinduism. The weaving of molecular strands within a discrete layer by anion-template metal-ion coordination opens the way for the synthesis of other molecular knot topologies and to woven polymer materials.

Protein corona, understanding the nanoparticle-protein interactions and future perspectives: A critical review

Proteins are biopolymers of highly varied structures taking part in almost all processes occurring in living cells. When nanoparticles (NPs) interact with proteins in biological environments, they are surrounded by a layer of biomolecules, mainly proteins adsorbing to the surfaces. This protein rich layer formed around NPs is called the "protein corona". Consequential interactions between NPs and proteins are governed due to the characteristics of the corona. The features of NPs such as the size, surface chemistry, charge are the critical factors influencing the behavior of protein corona. Molecular properties and protein corona composition affect the cellular uptake of NPs. Understanding and analyzing protein corona formation in relation to protein-NP properties, and elucidating its biological implications play an important role in bio-related nano-research studies. Protein-NP interactions have been studied extensively for the purpose of investigating the potential use of NPs as carriers in drug delivery systems. Further study should focus on exploring the effects of various characteristic parameters, such as the particle size, modifier type, temperature, pH on protein-NP interactions, providing toxicity information of novel NPs. In this contribution, important aspects related to protein corona forming, influential factors, novel findings and future perspectives on protein-NP interactions are overviewed.

Effects of turn-structure on folding and entanglement in artificial molecular overhand knots

The length and constitution of spacers linking three 2,6-pyridinedicarboxamide units in a molecular strand influence the tightness of the resulting overhand (open-trefoil) knot that the strand folds into in the presence of lanthanide(iii) ions. The use of β-hairpin forming motifs as linkers enables a metal-coordinated pseudopeptide with a knotted tertiary structure to be generated. The resulting pseudopeptide knot has one of the highest backbone-to-crossing ratios (BCR)—a measure of knot tightness (a high value corresponding to looseness)—for a synthetic molecular knot to date. Preorganization in the crossing-free turn section of the knot affects aromatic stacking interactions close to the crossing region. The metal-coordinated pseudopeptide knot is compared to overhand knots with other linkers of varying tightness and turn preorganization, and the entangled architectures characterized by NMR spectroscopy, ESI-MS, CD spectroscopy and, in one case, X-ray crystallography. The results show how it is possible to program specific conformational properties into different key regions of synthetic molecular knots, opening the way to systems where knotting can be systematically incorporated into peptide-like chains through design.

Spacers linking 2,6-pyridinedicarboxamide units influence the tightness of the corresponding lanthanide-coordinated overhand knot. β-Hairpin forming motifs generate a metal-coordinated pseudopeptide with a knotted tertiary structure.

Transmembrane Ion Channels Formed by a Star of David [2]Catenane and a Molecular Pentafoil Knot

A (Fe^II)₆-coordinated triply interlocked ("Star of David") [2]catenane (6₁² link) and a (Fe^II)₅-coordinated pentafoil (5₁) knot are found to selectively transport anions across phospholipid bilayers. Allostery, topology, and building block stoichiometry all play important roles in the efficacy of the ionophoric activity. Multiple Fe^II cation coordination by the interlocked molecules is crucial: the demetalated catenane exhibits no anion binding in solution nor any transmembrane ion transport properties. However, the topologically trivial, Lehn-type cyclic hexameric Fe^II helicates-which have similar anion binding affinities to the metalated Star of David catenane in solution-also display no ion transport properties. The unanticipated difference in behavior between the open- and closed-loop structures may arise from conformational restrictions in the linking groups that likely enhances the rigidity of the channel-forming topologically complex molecules. The (Fe^II)₆-coordinated Star of David catenane, derived from a hexameric cyclic helicate, is 2 orders of magnitude more potent in terms of ion transport than the (Fe^II)₅-coordinated pentafoil knot, derived from a cyclic pentamer of the same building block. The reduced efficacy is reminiscent of multisubunit protein ion channels assembled with incorrect monomer stoichiometries.

Isogonal weavings on the sphere: knots, links, polycatenanes

Mathematical knots and links are described as piecewise linear - straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'-transitive). Corner- and stick-transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.

Tying different knots in a molecular strand

The properties of knots are exploited in a range of applications, from shoelaces to the knots used for climbing, fishing and sailing¹. Although knots are found in DNA and proteins², and form randomly in other long polymer chains^3,4, methods for tying⁵ different sorts of knots in a synthetic nanoscale strand are lacking. Molecular knots of high symmetry have previously been synthesized by using non-covalent interactions to assemble and entangle molecular chains^6-15, but in such instances the template and/or strand structure intrinsically determines topology, which means that only one type of knot is usually possible. Here we show that interspersing coordination sites for different metal ions within an artificial molecular strand enables it to be tied into multiple knots. Three topoisomers-an unknot (0₁) macrocycle, a trefoil (3₁) knot^6-15, and a three-twist (5₂) knot-were each selectively prepared from the same molecular strand by using transition-metal and lanthanide ions to guide chain folding in a manner reminiscent of the action of protein chaperones¹⁶. We find that the metal-ion-induced folding can proceed with stereoinduction: in the case of one knot, a lanthanide(III)-coordinated crossing pattern formed only with a copper(I)-coordinated crossing of particular handedness. In an unanticipated finding, metal-ion coordination was also found to translocate an entanglement from one region of a knotted molecular structure to another, resulting in an increase in writhe (topological strain) in the new knotted conformation. The knot topology affects the chemical properties of the strand: whereas the tighter 5₂ knot can bind two different metal ions simultaneously, the looser 3₁ isomer can bind only either one copper(I) ion or one lutetium(III) ion. The ability to tie nanoscale chains into different knots offers opportunities to explore the modification of the structure and properties of synthetic oligomers, polymers and supramolecules.

Comparison of Fused Deposition Modeling and Color Jet 3D Printing Technologies for the Printing of Mathematical Geometries

Many mathematical geometries act as an optimal structure for functional applications and have always been an area of interest in the research field. Their topology offers properties which are crucial and can be used effectively in various domains. Apart from that, some have a resemblance to naturally occurring compounds which can help us to study their different transformations and behavior. In this paper, we present two such geometries, first, gyroid, which is an iso-minimal surface and second a three-crossing knot, also known as trefoil knot. The structure of gyroid makes it unique and is considered suitable in developing energy-absorbing, structural and lightweight applications. Similarly, some types of knots resemble the DNA structure and have found use in molecular chemistry. This paper discusses different application areas of these geometries. Further, this paper presents modeling and printing by using fused deposition modeling (FDM) and color jet printing (CJP). Comparative analysis has been done by considering various parameters. This paper discusses the potential of these two rapid prototyping technologies and their suitability for specific printing applications.

Protein knots provide mechano-resilience to an AAA+ protease-mediated proteolysis with profound ATP energy expenses

Knotted proteins are some of the most fascinating examples of how linear polypeptide chains can achieve intricate topological arrangements efficiently and spontaneously. The entanglements of polypeptide chains could potentially enhance their folding stabilities. We recently reported the unprecedented mechanostability of the Gordian (5₂) knotted family of human ubiquitin C-terminal hydrolases (UCHs) in the context of withstanding the mechanical unfolding of the bacterial AAA+ proteasome, ClpXP; a green fluorescence protein (GFP) was fused to the N-terminus of various UCHs as a reporter of the unfolding and degradation of these topologically knotted substrates, but it also limited the ability to examine the effect of untying the knotted topology via N-terminal truncation. In this study, we directly monitored the ClpXP-mediated degradation of UCH variants by electrophoresis and quantitative imaging analyses. We demonstrated that untying of the 5₂ knot in UCHL1 via N-terminal truncation (UCHL1_Δ11) significantly reduces its mechanostability. We further quantified the ATP expenditures of degrading different UCH variants by ClpXP. The unknotted UCHL1_Δ11 underwent accelerated ClpXP-dependent proteolysis, with a 30-fold reduction in ATP consumption compared to the knotted wild type. Unlike all other known ClpXP substrates, UCHL5, which is the most resilient substrate known to date, significantly slowed down the ATP turnover rate by ClpXP. Furthermore, UCHL5 required 1000-fold more ATP to be fully degraded by ClpXP compared to GFP. Our results underscored how the complex, knotted folding topology in UCHs may interfere with the mechano-unfolding processes of the AAA+ unfoldase, ClpX.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Evaluating the Jones polynomial with tensor networks

We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot is reduces to the partition function of a q-state anisotropic Potts model with complex interactions, which is defined on a planar signed graph that corresponds to the knot. For any integer q, we cast this partition function into tensor network form, which inherits the interaction graph structure of the Potts model instance, and employ fast tensor network contraction protocols to obtain the exact tensor trace and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace exactly, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.