Knotting and unknotting of a protein in single molecule experiments

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.

Topological Reaction Coordinate Captures the Folding Transition State Ensemble in a Pierced Lasso Protein

Proteins with a pierced lasso topology (PLT) have a covalent loop created by a disulfide bond, and the backbone circles back to thread the loop. This threaded topology has unique features compared to knotted topologies; notably, the topology is controlled by the chemical environment and the covalent loop remains intact even when denatured. In this work, we use the hormone leptin as our model PLT system and study its folding using molecular dynamics simulations that employ a structure-based (Go̅-like) model. We find that the reduced protein has a two-state folding mechanism with a transition state ensemble (TSE) that can be characterized by the reaction coordinate Q, the fraction of native contacts formed. In contrast, the oxidized protein, which must thread part of the polypeptide chain through a covalent loop, has a folding process that is poorly characterized by Q. Instead, we find that a topological coordinate that monitors the residue crossing the loop can identify the TSE of oxidized leptin. By precisely identifying the predicted TSE, one may now reliably calculate theoretical phi-values for the PLT protein, thereby enabling a comparison with experimental measurements. We find the loop-threading constraint leads to noncanonical phi-values that are uniformly small because this PLT protein has a flat energy landscape through the TSE.

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.

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.

De novo design of knotted tandem repeat proteins

De novo protein design methods can create proteins with folds not yet seen in nature. These methods largely focus on optimizing the compatibility between the designed sequence and the intended conformation, without explicit consideration of protein folding pathways. Deeply knotted proteins, whose topologies may introduce substantial barriers to folding, thus represent an interesting test case for protein design. Here we report our attempts to design proteins with trefoil (31) and pentafoil (51) knotted topologies. We extended previously described algorithms for tandem repeat protein design in order to construct deeply knotted backbones and matching designed repeat sequences (N = 3 repeats for the trefoil and N = 5 for the pentafoil). We confirmed the intended conformation for the trefoil design by X ray crystallography, and we report here on this protein's structure, stability, and folding behaviour. The pentafoil design misfolded into an asymmetric structure (despite a 5-fold symmetric sequence); two of the four repeat-repeat units matched the designed backbone while the other two diverged to form local contacts, leading to a trefoil rather than pentafoil knotted topology. Our results also provide insights into the folding of knotted proteins.

Effect of temperature and nucleotide on the binding of BiP chaperone to a protein substrate

BiP (immunoglobulin heavy-chain binding protein) is a Hsp70 monomeric ATPase motor that plays broad and crucial roles in maintaining proteostasis inside the cell. Structurally, BiP is formed by two domains, a nucleotide-binding domain (NBD) with ATPase activity connected by a flexible hydrophobic linker to the substrate-binding domain. While the ATPase and substrate binding activities of BiP are allosterically coupled, the latter is also dependent on nucleotide binding. Recent structural studies have provided new insights into BiP's allostery; however, the influence of temperature on the coupling between substrate and nucleotide binding to BiP remains unexplored. Here, we study BiP's binding to its substrate at the single molecule level using thermo-regulated optical tweezers which allows us to mechanically unfold the client protein and explore the effect of temperature and different nucleotides on BiP binding. Our results confirm that the affinity of BiP for its protein substrate relies on nucleotide binding, by mainly regulating the binding kinetics between BiP and its substrate. Interestingly, our findings also showed that the apparent affinity of BiP for its protein substrate in the presence of nucleotides remains invariable over a wide range of temperatures, suggesting that BiP may interact with its client proteins with similar affinities even when the temperature is not optimal. Thus, BiP could play a role as a "thermal buffer" in proteostasis.

Temperature dependent mechanical unfolding and refolding of a protein studied by thermo-regulated optical tweezers

Temperature is a useful system variable to gather kinetic and thermodynamic information from proteins. Usually, free energy and the associated entropic and enthalpic contributions are obtained by quantifying the conformational equilibrium based on melting experiments performed in bulk conditions. Such experiments are suitable only for those small single-domain proteins whose side reactions of irreversible aggregation are unlikely to occur. Here, we avoid aggregation by pulling single-protein molecules in a thermo-regulated optical tweezers. Thus, we are able to explore the temperature dependence of the thermodynamic and kinetic parameters of MJ0366 from Methanocaldococcus jannaschii at the single-molecule level. By performing force-ramp experiments between 2°C and 40°C, we found that MJ0366 has a nonlinear dependence of free energy with temperature and a specific heat change of 2.3 ± 1.2 kcal/mol∗K. These thermodynamic parameters are compatible with a two-state unfolding/refolding mechanism for MJ0366. However, the kinetics measured as a function of the temperature show a complex behavior, suggesting a three-state folding mechanism comprising a high-energy intermediate state. The combination of two perturbations, temperature and force, reveals a high-energy species in the folding mechanism of MJ0366 not detected in force-ramp experiments at constant temperature.

Stochastic model of translocation of knotted proteins

Knotted proteins, when forced through the pores, can get stuck if the knots in their backbone tighten under force. Alternatively, the knot can slide off the chain, making translocation possible. We construct a simple energy landscape model of this process with a time-periodic potential that mimics the action of a molecular motor. We calculate the translocation time as a function of the period of the pulling force, discuss the asymptotic limits and biological relevance of the results.

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.

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.

Revealing Topological Barriers against Knot Untying in Thermal and Mechanical Protein Unfolding by Molecular Dynamics Simulations

The knot is one of the most remarkable topological features identified in an increasing number of proteins with important functions. However, little is known about how the knot is formed during protein folding, and untied or maintained in protein unfolding. By means of all-atom molecular dynamics simulation, here we employ methyltransferase YbeA as the knotted protein model to analyze changes of the knotted conformation coupled with protein unfolding under thermal and mechanical denaturing conditions. Our results show that the trefoil knot in YbeA is occasionally untied via knot loosening rather than sliding under enhanced thermal fluctuations. Through correlating protein unfolding with changes in the knot position and size, several aspects of barriers that jointly suppress knot untying are revealed. In particular, protein unfolding is always prior to knot untying and starts preferentially from separation of two α-helices (α1 and α5), which protect the hydrophobic core consisting of β-sheets (β1–β4) from exposure to water. These β-sheets form a loop through which α5 is threaded to form the knot. Hydrophobic and hydrogen bonding interactions inside the core stabilize the loop against loosening. In addition, residues at N-terminal of α5 define a rigid turning to impede α5 from sliding out of the loop. Site mutations are designed to specifically eliminate these barriers, and easier knot untying is achieved under the same denaturing conditions. These results provide new molecular level insights into the folding/unfolding of knotted proteins.

Topological and physical links in soft matter systems

Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packedin vivoto interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.

Slipknotted and unknotted monovalent cation-proton antiporters evolved from a common ancestor

While the slipknot topology in proteins has been known for over a decade, its evolutionary origin is still a mystery. We have identified a previously overlooked slipknot motif in a family of two-domain membrane transporters. Moreover, we found that these proteins are homologous to several families of unknotted membrane proteins. This allows us to directly investigate the evolution of the slipknot motif. Based on our comprehensive analysis of 17 distantly related protein families, we have found that slipknotted and unknotted proteins share a common structural motif. Furthermore, this motif is conserved on the sequential level as well. Our results suggest that, regardless of topology, the proteins we studied evolved from a common unknotted ancestor single domain protein. Our phylogenetic analysis suggests the presence of at least seven parallel evolutionary scenarios that led to the current diversity of proteins in question. The tools we have developed in the process can now be used to investigate the evolution of other repeated-domain proteins.

In proteins with the slipknot topology, the polypeptide chain forms a slipknot—a structure that is not necessarily manifest to a naked eye, but it can be detected using mathematical methods. Slipknots are conserved motifs often found at catalytic sites and are directly involved in molecular transport. Although the first proteins with slipknots were found in 2007, many questions remain unanswered, e.g. how these proteins appeared, or whether the slipknotted proteins evolved from unknotted ones or vice versa. Here we provide the first analysis of homologous slipknotted and unknotted transmembrane proteins in order to elucidate their evolutionary relationship. We show that two-domain slipknotted and unknotted membrane transporters share the same one-domain unknotted protein as an ancestor. The ancestor gene duplicated and underwent various diversification and fusion events during the evolution, which have led to the appearance of a large superfamily of secondary active transporters. The slipknot motif seems to have been created by chance after a fusion of two single domain genes. Therefore, we show here that the slipknotted transporter evolved from an unknotted one-domain protein and that there are at least seven different evolutionary scenarios that gave rise to this large superfamily of transporters.

Single-Molecule Force Spectroscopy of Protein Folding

The use of force probes to induce unfolding and refolding of single molecules through the application of mechanical tension, known as single-molecule force spectroscopy (SMFS), has proven to be a powerful tool for studying the dynamics of protein folding. Here we provide an overview of what has been learned about protein folding using SMFS, from small, single-domain proteins to large, multi-domain proteins. We highlight the ability of SMFS to measure the energy landscapes underlying folding, to map complex pathways for native and non-native folding, to probe the mechanisms of chaperones that assist with native folding, to elucidate the effects of the ribosome on co-translational folding, and to monitor the folding of membrane proteins.

Model systems for optical trapping: the physical basis and biological applications

The micromechanical methods, among which optical trapping and atomic force microscopy have a special place, are widespread currently in biology to study molecular interactions between different biological objects. Optical trapping is reported to be quite applicable to study the mechanical properties of surface structures onto bacterial (pili and flagella) and eukaryotic (filopodia) cells. The review briefly summarizes the physical basis of optical trapping, as well as the principles of calculating the van der Waals, electrostatic, and donor-acceptor forces when two microparticles or a microparticle and a flat surface are used. Three main types of model systems (abiotic, biotic, and mixed) used in trapping experiments are described, and the peculiarities of manipulation with living (bacteria, fungal spores, etc.) and non-spherical objects (e.g., rod-shaped bacteria) are summarized.

Unfolding and Translocation of Knotted Proteins by Clp Biological Nanomachines: Synergistic Contribution of Primary Sequence and Topology Revealed by Molecular Dynamics Simulations

We use Langevin dynamics simulations to model, at an atomistic resolution, how various natively knotted proteins are unfolded in repeated allosteric translocating cycles of the ClpY ATPase. We consider proteins representative of different topologies, from the simplest knot (trefoil 3₁), to the three-twist 5₂ knot, to the most complex stevedore, 6₁, knot. We harness the atomistic detail of the simulations to address aspects that have so far remained largely unexplored, such as sequence-dependent effects on the ruggedness of the landscape traversed during knot sliding. Our simulations reveal the combined effect on translocation of the knotted protein structure, i.e., backbone topology and geometry, and primary sequence, i.e., side chain size and interactions, and show that the latter can dominate translocation hindrance. In addition, we observe that due to the interplay between the knotted topology and intramolecular contacts the transmission of tension along the polypeptide chain occurs very differently from that of homopolymers. Finally, by considering native and non-native interactions, we examine how the disruption or formation of such contacts can affect the translocation processivity and concomitantly create multiple unfolding pathways with very different activation barriers.

Mechanical strength of RNA knot in Zika virus protects against cellular defenses

Unusual knot-like structures recently discovered in viral exoribonuclease-resistant RNAs (xrRNAs) prevent digestion by host RNases to create subgenomic RNAs enhancing infection and pathogenicity. xrRNAs are proposed to prevent digestion through mechanical resistance to unfolding. However, their unfolding force has not been measured, and the factors determining RNase resistance are unclear. Furthermore, how these knots fold remains unknown. Unfolding a Zika virus xrRNA with optical tweezers revealed that it was the most mechanically stable RNA yet observed. The knot formed by threading the 5' end into a three-helix junction before pseudoknot interactions closed a ring around it. The pseudoknot and tertiary contacts stabilizing the threaded 5' end were both required to generate extreme force resistance, whereas removing a 5'-end contact produced a low-force knot lacking RNase resistance. These results indicate mechanical resistance plays a central functional role, with the fraction of molecules forming extremely high-force knots determining the RNase resistance level.

A Specific Set of Heterogeneous Native Interactions Yields Efficient Knotting in Protein Folding

Native interactions are crucial for folding, and non-native interactions appear to be critical for efficiently knotting proteins. Therefore, it is important to understand both their roles in the folding of knotted proteins. It has been proposed that non-native interactions drive the correct order of contact formation, which is essential to avoid backtracking and efficiently self-tie. In this study, we ask if non-native interactions are strictly necessary to tangle a protein or if the correct order of contact formation can be assured by a specific set of native, but otherwise heterogeneous (i.e., having distinct energies), interactions. In order to address this problem, we conducted extensive Monte Carlo simulations of lattice models of protein-like sequences designed to fold into a preselected knotted conformation embedding a trefoil knot. We were able to identify a specific set of heterogeneous native interactions that drives efficient knotting and is able to fold the protein when combined with the remaining native interactions modeled as homogeneous. This specific set of heterogeneous native interactions is strictly enough to efficiently self-tie. A distinctive feature of these native interactions is that they do not backtrack because their energies ensure the correct order of contact formation. Furthermore, they stabilize a knotted intermediate state, which is en route to the native structure. Our results thus show that-at least in the context of the adopted model-non-native interactions are not necessary to knot a protein. However, when they are taken into account in protein energetics, it is possible to find specific, nonlocal non-native interactions that operate as a scaffold that assists the knotting step.

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.

DNA Knot Malleability in Single-Digit Nanopores

Knots in long DNA molecules are prevalent in biological systems and serve as a model system for investigating static and dynamic properties of biopolymers. We explore the dynamics of knots in double-stranded DNA in a new regime of nanometer-scale confinement, large forces, and short time scales, using solid-state nanopores. We show that DNA knots undergo isomorphic translocation through a nanopore, retaining their equilibrium morphology by swiftly compressing in a lateral direction to fit the constriction. We observe no evidence of knot tightening or jamming, even for single-digit nanopores. We explain the observations as the malleability of DNA, characterized by sharp buckling of the DNA in nanopores, driven by the transient disruption of base pairing. Our molecular dynamics simulations support the model. These results are relevant not only for the understanding of DNA packing and manipulation in living cells but also for the polymer physics of DNA and the development of nanopore-based sequencing technologies.

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.

Mechanically tightening, untying and retying a protein trefoil knot by single-molecule force spectroscopy

Knotted conformation is one of the most surprising topological features found in proteins, and understanding the folding mechanism of such knotted proteins remains a challenge. Here, we used optical tweezers (OT) to investigate the mechanical unfolding and folding behavior of a knotted protein Escherichia coli tRNA (guanosine-1) methyltransferase (TrmD). We found that when stretched from its N- and C-termini, TrmD can be mechanically unfolded and stretched into a tightened trefoil knot, which is composed of ca. 17 residues. Stretching of the unfolded TrmD involved a compaction process of the trefoil knot at low forces. The unfolding pathways of the TrmD were bifurcated, involving two-state and three-state pathways. Upon relaxation, the tightened trefoil knot loosened up first, leading to the expansion of the knot, and the unfolded TrmD can then fold back to its native state efficiently. By using an engineered truncation TrmD variant, we stretched TrmD along a pulling direction to allow us to mechanically unfold TrmD and untie the trefoil knot. We found that the folding of TrmD from its unfolded polypeptide without the knot is significantly slower. The knotting is the rate-limiting step of the folding of TrmD. Our results highlighted the critical importance of the knot conformation for the folding and stability of TrmD, offering a new perspective to understand the role of the trefoil knot in the biological function of TrmD.

Optical tweezers are used to stretch a knotted protein along different directions to probe its unfolding–folding behaviors, and the conformational change of its knot structure.

Generalized Circuit Topology of Folded Linear Chains

A wide range of physical systems can be formally mapped to a linear chain of sorted objects. Upon introduction of intrachain interactions, such a chain can "fold" to elaborate topological structures, analogous to folded linear polymer systems. Two distinct chain-topology theories, knot theory and circuit topology, have separately provided insight into the structure, dynamics, and evolution of folded linear polymers such as proteins and genomic DNA. Knot theory, however, ignores intrachain interactions (contacts), whereas chain crossings are ignored in circuit topology. Thus, there is a need for a universal approach that can provide topological description of any folded linear chain. Here, we generalize circuit topology in order to grasp particularities typically addressed by knot theory. We develop a generic approach that is simple, mathematically rigorous, and practically useful for structural classification, analysis of structural dynamics, and engineering applications.

Directional translocation resistance of Zika xrRNA

xrRNAs from flaviviruses survive in host cells because of their exceptional dichotomic response to the unfolding action of different enzymes. They can be unwound, and hence copied, by replicases, and yet can resist degradation by exonucleases. How the same stretch of xrRNA can encode such diverse responses is an open question. Here, by using atomistic models and translocation simulations, we uncover an elaborate and directional mechanism for how stress propagates when the two xrRNA ends, [Formula: see text] and [Formula: see text], are driven through a pore. Pulling the [Formula: see text] end, as done by replicases, elicits a progressive unfolding; pulling the [Formula: see text] end, as done by exonucleases, triggers a counterintuitive molecular tightening. Thus, in what appears to be a remarkable instance of intra-molecular tensegrity, the very pulling of the [Formula: see text] end is what boosts resistance to translocation and consequently to degradation. The uncovered mechanistic principle might be co-opted to design molecular meta-materials.

Mechanical unfolding of a knotted protein unveils the kinetic and thermodynamic consequences of threading a polypeptide chain

Knots are remarkable topological features in nature. The presence of knots in crystallographic structures of proteins have stimulated considerable research to determine the kinetic and thermodynamic consequences of threading a polypeptide chain. By mechanically manipulating MJ0366, a small single domain protein harboring a shallow trefoil knot, we allow the protein to refold from either the knotted or the unknotted denatured state to characterize the free energy profile associated to both folding pathways. By comparing the stability of the native state with reference to the knotted and unknotted denatured state we find that knotting the polypeptide chain of MJ0366 increase the folding energy barrier in a magnitude close to the energy cost of forming a knot randomly in the denatured state. These results support that a protein knot can be formed during a single cooperative step of folding but occurs at the expenses of a large increment on the free energy barrier.

Restriction of S-adenosylmethionine conformational freedom by knotted protein binding sites

S-adenosylmethionine (SAM) is one of the most important enzyme substrates. It is vital for the function of various proteins, including large group of methyltransferases (MTs). Intriguingly, some bacterial and eukaryotic MTs, while catalysing the same reaction, possess significantly different topologies, with the former being a knotted one. Here, we conducted a comprehensive analysis of SAM conformational space and factors that affect its vastness. We investigated SAM in two forms: free in water (via NMR studies and explicit solvent simulations) and bound to proteins (based on all data available in the PDB and on all-atom molecular dynamics simulations in water). We identified structural descriptors—angles which show the major differences in SAM conformation between unknotted and knotted methyltransferases. Moreover, we report that this is caused mainly by a characteristic for knotted MTs compact binding site formed by the knot and the presence of adenine-binding loop. Additionally, we elucidate conformational restrictions imposed on SAM molecules by other protein groups in comparison to conformational space in water.

The topology of a folded polypeptide chain has great impact on the resulting protein function and its interaction with ligands. Interestingly, topological constraints appear to affect binding of one of the most ubiquitous substrates in the cell, S-adenosylmethionine (SAM), to its target proteins. Here, we demonstrate how binding sites of specific proteins restrict SAM conformational freedom in comparison to its unbound state, with a special interest in proteins with non-trivial topology, including an exciting group of knotted methyltransferases. Using a vast array of computational methods combined with NMR experiments, we identify key structural features of knotted methyltransferases that impose unorthodox SAM conformations. We compare them with the characteristics of standard, unknotted SAM binding proteins. These results are significant for understanding differences between analogous, yet topologically different enzymes, as well as for future rational drug design.

Force Spectroscopy on Single Molecules of Life

Biomolecules such as nucleic acids and proteins constitute the cells and its organelles that form the crucial components in all living organisms. They are associated with a variety of cellular processes during which they undergo conformational orientations. The structural rearrangements resulting from protein-protein, protein-DNA, and protein-drug interactions vary in spatial and temporal length scales. Force is one of the important key factors which regulate these interactions. The magnitude of the force can vary from sub-piconewtons to several thousands of piconewtons. Single-molecule force spectroscopy acts as a powerful tool which is capable of investigating mechanical stability and conformational rearrangements arising in biomolecules due to the above interactions. Real-time observation of conformational dynamics including access to rare or transient states and the estimation of mean dwell times using these tools aids in the kinetic analysis of these interactions. In this review, we highlight the capabilities of common force spectroscopy techniques such as optical tweezers, magnetic tweezers, and atomic force microscopy with case studies on emerging applications.

On folding of entangled proteins: knots, lassos, links and θ-curves

Around 6% of protein structures deposited in the PDB are entangled, forming knots, slipknots, lassos, links, and θ-curves. In each of these cases, the protein backbone weaves through itself in a complex way, and at some point passes through a closed loop, formed by other regions of the protein structure. Such a passing can be interpreted as crossing a topological barrier. How proteins overcome such barriers, and therefore different degrees of frustration, challenged scientists and has shed new light on the field of protein folding. In this review, we summarize the current knowledge about the free energy landscape of proteins with non-trivial topology. We describe identified mechanisms which lead proteins to self-tying. We discuss the influence of excluded volume, such as crowding and chaperones, on tying, based on available data. We briefly discuss the diversity of topological complexity of proteins and their evolution. We also list available tools to investigate non-trivial topology. Finally, we formulate intriguing and challenging questions at the boundary of biophysics, bioinformatics, biology, and mathematics, which arise from the discovery of entangled proteins.

s-Block metal ions induce structural transformations between figure-eight and double trefoil knots

The presence or absence of s-block metal ions induces reversible structural transformation of molecular knots.

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.

Revealing Cooperation between Knotted Conformation and Dimerization in Protein Stabilization by Molecular Dynamics Simulations

The topological knot is thought to play a stabilizing role in maintaining the global fold and nature of proteins with the underlying mechanism yet to be elucidated. Given that most proteins containing trefoil knots exist and function as homodimers with a large part of the dimer interface occupied by the knotted region, we reason that the knotted conformation cooperates with dimerization in protein stabilization. Here, we take YbeA from Escherichia coli as the knotted protein model, using molecular dynamics (MD) simulations to compare the stability of two pairs of dimeric proteins having the same sequence and secondary structures but differing in the presence or absence of a trefoil knot in each subunit. The dimer interface of YbeA is identified to involve favorable contacts among three α-helices (α1, α3, and α5), one of which (α5) is threaded through a loop connected with α3 to form the knot. Upon removal of the knot by appropriate change of the knot-making crossing of the polypeptide chain, relevant domains are less constrained and exhibit enhanced fluctuations to decrease contacts at the interface. Unknotted subunits are less compact and undergo structural changes to ease the dimer separation. Such a stabilizing effect is evidenced by steered MD simulations, showing that the mechanical force required for dimer separation is significantly reduced by removing the knot. In addition to the knotted conformation, dimerization further improves the protein stability by restricting the α1-α5 separation, which is defined as a leading step for protein unfolding. These results provide important insights into the structure-function relationship of dimerization in knotted proteins.

Conformational State Hopping of Knots in Tensioned Polymer Chains

We use Brownian dynamics simulations to study the conformational states of knots on tensioned chains. Focusing specifically on the 8₁ knot, we observe knot conformational state hopping and show that the process can be described by a two-state kinetic model in the presence of an external force. The distribution of knot conformational states depends on the applied chain tension, which leads to a force-dependent distribution of knot untying pathways. We generalize our findings by considering the untying pathways of other knots and find that the way knots untie is generally governed by the force applied to the chain. From a broader perspective, being able to influence how a knot unties via external force can potentially be useful for applications of single-molecule techniques in which knots are unwanted.

Interactive molecular dynamics in virtual reality from quantum chemistry to drug binding: An open-source multi-person framework

As molecular scientists have made progress in their ability to engineer nanoscale molecular structure, we face new challenges in our ability to engineer molecular dynamics (MD) and flexibility. Dynamics at the molecular scale differs from the familiar mechanics of everyday objects because it involves a complicated, highly correlated, and three-dimensional many-body dynamical choreography which is often nonintuitive even for highly trained researchers. We recently described how interactive molecular dynamics in virtual reality (iMD-VR) can help to meet this challenge, enabling researchers to manipulate real-time MD simulations of flexible structures in 3D. In this article, we outline various efforts to extend immersive technologies to the molecular sciences, and we introduce "Narupa," a flexible, open-source, multiperson iMD-VR software framework which enables groups of researchers to simultaneously cohabit real-time simulation environments to interactively visualize and manipulate the dynamics of molecular structures with atomic-level precision. We outline several application domains where iMD-VR is facilitating research, communication, and creative approaches within the molecular sciences, including training machines to learn potential energy functions, biomolecular conformational sampling, protein-ligand binding, reaction discovery using "on-the-fly" quantum chemistry, and transport dynamics in materials. We touch on iMD-VR's various cognitive and perceptual affordances and outline how these provide research insight for molecular systems. By synergistically combining human spatial reasoning and design insight with computational automation, technologies such as iMD-VR have the potential to improve our ability to understand, engineer, and communicate microscopic dynamical behavior, offering the potential to usher in a new paradigm for engineering molecules and nano-architectures.

Hydrophobic confinement modulates thermal stability and assists knotting in the folding of tangled proteins

There is growing support for the idea that the in vivo folding process of knotted proteins is assisted by chaperonins, but the mechanism of chaperonin assisted folding remains elusive. Here, we conduct extensive Monte Carlo simulations of lattice and off-lattice models to explore the effects of confinement and hydrophobic intermolecular interactions with the chaperonin cage in the folding and knotting processes. We find that moderate to high protein-cavity interactions (which are likely to be established in the beginning of the chaperonin working cycle) cause an energetic destabilization of the protein that overcomes the entropic stabilization driven by excluded volume, and leads to a decrease of the melting temperature relative to bulk conditions. Moreover, mild-to-moderate hydrophobic interactions with the cavity (which would be established later in the cycle) lead to a significant enhancement of knotting probability in relation to bulk conditions while simultaneously moderating the effect of steric confinement in the enhancement of thermal stability. Our results thus indicate that the chaperonin may be able to assist knotting without simultaneously thermally stabilizing potential misfolded states to a point that would hamper productive folding thus compromising its functional role.

Sequence and structural patterns detected in entangled proteins reveal the importance of co-translational folding

Proteins must fold quickly to acquire their biologically functional three-dimensional native structures. Hence, these are mainly stabilized by local contacts, while intricate topologies such as knots are rare. Here, we reveal the existence of specific patterns adopted by protein sequences and structures to deal with backbone self-entanglement. A large scale analysis of the Protein Data Bank shows that loops significantly intertwined with another chain portion are typically closed by weakly bound amino acids. Why is this energetic frustration maintained? A possible picture is that entangled loops are formed only toward the end of the folding process to avoid kinetic traps. Consistently, these loops are more frequently found to be wrapped around a portion of the chain on their N-terminal side, the one translated earlier at the ribosome. Finally, these motifs are less abundant in natural native states than in simulated protein-like structures, yet they appear in 32% of proteins, which in some cases display an amazingly complex intertwining.

Direct observation of the fast and robust folding of a slipknotted protein by optical tweezers

Understanding the folding mechanism of knotted and slipknotted proteins has attracted considerable interest. Due to their topological complexity, knotted and slipknotted proteins are predicted to fold slowly and involve large topological barriers. Molecular dynamics simulation studies suggest that a slipknotted conformation can serve as an important intermediate to help greatly reduce the topological difficulty during the folding of some knotted proteins. Here we use a single molecule optical tweezers technique to directly probe the folding of a small slipknotted protein AFV3-109. We found that stretching AFV3-109 can lead to the untying of the slipknot and complete unfolding of AFV3-109. Upon relaxation, AFV3-109 can readily refold back to its native slipknot conformation with high fidelity when the stretching force is lower than 6 pN. The refolding of AFV3-109 occurs in a sharp two-state like transition. Our results indicate that, different from knotted proteins, the folding of a slipknotted protein like AFV3-109 can be fast, and may not necessarily involve a high topological barrier.

The AAA+ protease ClpXP can easily degrade a 31 and a 52-knotted protein

Knots in proteins are hypothesized to make them resistant to enzymatic degradation by ATP-dependent proteases and recent studies have shown that whereas ClpXP can easily degrade a protein with a shallow 3₁ knot, it cannot degrade 5₂-knotted proteins if degradation is initiated at the C-terminus. Here, we present detailed studies of the degradation of both 3₁- and 5₂-knotted proteins by ClpXP using numerous constructs where proteins are tagged for degradation at both N- and C-termini. Our results confirm and extend earlier work and show that ClpXP can easily degrade a deeply 3₁-knotted protein. In contrast to recently published work on the degradation of 5₂-knotted proteins, our results show that the ClpXP machinery can also easily degrade these proteins. However, the degradation depends critically on the location of the degradation tag and the local stability near the tag. Our results are consistent with mechanisms in which either the knot simply slips along the polypeptide chain and falls off the free terminus, or one in which the tightened knot enters the translocation pore of ClpXP. Results of experiments on knotted protein fusions with a highly stable domain show partial degradation and the formation of degradation intermediates.

Effects of knot tightness at the molecular level

Three 819 knots in closed-loop strands of different lengths (∼20, 23, and 26 nm) were used to experimentally assess the consequences of knot tightness at the molecular level. Through the use of 1H NMR, diffusion-ordered spectroscopy (DOSY), circular dichroism (CD), collision-induced dissociation mass spectrometry (CID-MS) and molecular dynamics (MD) simulations on the different-sized knots, we find that the structure, dynamics, and reactivity of the molecular chains are dramatically affected by the tightness of the knotting. The tautness of entanglement causes differences in conformation, enhances the expression of topological chirality, weakens covalent bonds, inhibits decomplexation events, and changes absorption properties. Understanding the effects of tightening nanoscale knots may usefully inform the design of knotted and entangled molecular materials.

Two-step self-assembly of a spider silk molecular clamp

Web spiders synthesize silk fibers of unique strength and extensibility through the controlled self-assembly of protein building blocks, so-called spidroins. The spidroin C-terminal domain is highly conserved and connects two polypeptide chains through formation of an all-helical, intertwined dimer. Here we use contact-induced fluorescence self-quenching and resonance energy transfer in combination with far-UV circular dichroism spectroscopy as three orthogonal structural probes to dissect the mechanism of folding and dimerization of a spidroin C-terminal domain from the major ampullate gland of the nursery web spider Euprosthenops australis. We show that helices forming the dimer core assemble very rapidly and fold on association. Subsequently, peripheral helices fold and dock slowly onto the preformed core. Lability of outer helices facilitates formation of a highly expanded, partially folded dimer. The high end-to-end distance of chain termini in the partially folded dimer suggests an extensibility module that contributes to elasticity of spider silk.

Stabilizing Effect of Inherent Knots on Proteins Revealed by Molecular Dynamics Simulations

A growing number of proteins have been identified as knotted in their native structures, with such entangled topological features being expected to play stabilizing roles maintaining both the global fold and the nature of proteins. However, the molecular mechanism underlying the stabilizing effect is ambiguous. Here, we combine unbiased and mechanical atomistic molecular dynamics simulations to investigate how a protein is stabilized by an inherent knot by directly comparing chemical, thermal, and mechanical denaturing properties of two proteins having the same sequence and secondary structures but differing in the presence or absence of an inherent knot. One protein is YbeA from Escherichia coli, containing a deep trefoil knot within the sequence, and the other is the modified protein with the knot of YbeA being removed. Under certain chemical denaturing conditions, the unknotted protein fully unfolds whereas the knotted protein does not, suggesting a higher intrinsic stability for the protein having a knot. Both proteins unfold under enhanced thermal fluctuations but at different rates and with distinct pathways. Opening the hydrophobic core via separation between two α-helices is identified as a crucial step initiating the protein unfolding, which, however, is restrained for the knotted protein by topological and geometrical frustrations. Energy barriers for denaturing the protein are reduced by removing the knot, as evidenced by mechanical unfolding simulations. Finally, yet importantly, no obvious change in size or location of the knot was observed during denaturing processes, indicating that YbeA may remain knotted for a relatively long time during and after denaturation.

Proteins' Knotty Problems

Knots in proteins are increasingly being recognized as an important structural concept, and the folding of these peculiar structures still poses considerable challenges. From a functional point of view, most protein knots discovered so far are either enzymes or DNA-binding proteins. Our comprehensive topological analysis of the Protein Data Bank reveals several novel structures including knotted mitochondrial proteins and the most deeply embedded protein knot discovered so far. For the latter, we propose a novel folding pathway based on the idea that a loose knot forms at a terminus and slides to its native position. For the mitochondrial proteins, we discuss the folding problem from the perspective of transport and suggest that they fold inside the mitochondria. We also discuss the evolutionary origin of a novel class of knotted membrane proteins and argue that a novel knotted DNA-binding protein constitutes a new fold. Finally, we have also discovered a knot in an artificially designed protein structure.

The Pathways of the iRFP713 Unfolding Induced by Different Denaturants

Near-infrared fluorescent proteins (NIR FPs) based on the complexes of bacterial phytochromes with their natural biliverdin chromophore are widely used as genetically encoded optical probes for visualization of cellular processes and deep-tissue imaging of cells and organs in living animals. In this work, we show that the steady-state and kinetic dependencies of the various spectral characteristics of iRFP713, developed from the bacterial phytochrome RpBphP2 and recorded at protein unfolding induced by guanidine hydrochloride (GdnHCl), guanidine thiocyanate (GTC), and urea, differ substantially. A study of the unfolding of three single-tryptophan mutant forms of iRFP713 expectedly revealed that protein unfolding begins with the dissociation of the native dimer, while the monomers remain compact. A further increase in the denaturant concentration leads to the formation of an intermediate state of iRFP713 having hydrophobic areas exposed on the protein surface (I). The total surface charge of iRFP713 (pI 5.86) changes from negative to positive with an increase in the concentration of GdnHCl and GTC because the negative charge of glutamic and aspartic acids is neutralized by forming salt bridges between the carboxyl groups and GdnH⁺ ions and because the guanidinium cations bind to amide groups of glutamines and asparagines. The coincidence of both the concentration of the denaturants at which the intermediate state of iRFP713 accumulates and the concentration of GdnH⁺ ions at which the neutralization of the surface charge of the protein in this state is ensured results in strong protein aggregation. This is evidently realized by iRFP713 unfolding by GTC. At the unfolding of the protein by GdnHCl, an intermediate state is populated at higher denaturant concentrations and a strong aggregation is not observed. As expected, protein aggregates are not formed in the presence of the urea. The aggregation of the protein upon neutralization of the charge on the macromolecule surface is the main indicator of the intermediate state of protein. The unfolded state of iRFP713, whose formation is accompanied by a significant decrease in the parameter A, was found to have a different residual structure in the denaturants used.