Pierced Lasso Bundles are a new class of knot-like motifs

Folds from fold: Exploring topological isoforms of a single-domain protein

Expanding the protein fold space beyond linear chains is of fundamental significance, yet remains largely unexplored. Herein, we report the creation of seven topological isoforms (i.e., linear, cyclic, knot, lasso, pseudorotaxane, and catenane) from a single protein fold precursor by rewiring the connectivity of secondary structure elements of the SpyTag-SpyCatcher complex and mutating the reactive residue on SpyTag to abolish the isopeptide bonding. These topological isoforms can be directly expressed in cells. Their topologies were confirmed by combined techniques of proteolytic digestion, fluorescence correlation spectroscopy (FCS), size-exclusion chromatography (SEC), and topological transformation. To study the effects of topology on their structures and properties, their biophysical properties were characterized by differential scanning calorimetry (DSC), heteronuclear single quantum coherence nuclear magnetic resonance spectroscopy (HSQC-NMR), and circular dichroism (CD) spectroscopy. Molecular dynamics (MD) simulations were further performed to reveal the atomic details of structural changes upon unfolding. Both experimental and simulation results suggest that they share a similar, well-folded hydrophobic core but exhibit distinct folding/unfolding dynamic behaviors. These results shed light onto the folding landscape of topological isoforms derived from the same protein fold. As a model system, this work improves our understanding of protein structure and dynamics beyond linear chains and suggests that protein folds are highly amenable to topological variation.

Exploring the folding landscape of leptin: Insights into threading pathways

The discovery of new protein topologies with entanglements and loop-crossings have shown the impact of local amino acid arrangement and global three-dimensional structures. This phenomenon plays a crucial role in understanding how protein structure relates to folding and function, affecting the global stability, and biological activity. Protein entanglements encompassing knots and non-trivial topologies add complexity to their folding free energy landscapes. However, the initial native contacts driving the threading event for entangled proteins remains elusive. The Pierced Lasso Topology (PLT) represents an entangled topology where a covalent linker creates a loop in which the polypeptide backbone is threaded through. Compared to true knotted topologies, PLTs are simpler topologies where the covalent-loop persists in all conformations. In this work, the PLT protein leptin, is used to visualize and differentiate the preference for slipknotting over plugging transition pathways along the folding route. We utilize the Energy Landscape Visualization Method (ELViM), a multidimensional projection technique, to visualize and distinguish early threaded conformations that cannot be observed in an in vitro experiment. Critical contacts for the leptin threading mechanisms were identified where the competing pathways are determined by the formation of a hairpin loop in the unfolded basin. Thus, prohibiting the dominant slipknotting pathway. Furthermore, ELViM offers insights into distinct folding pathways associated with slipknotting and plugging providing a novel tool for de novo design and in vitro experiments with residue specific information of threading events in silico.

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.

Dynamic Counterion Condensation Model Decodes Functional Dynamics of RNA Pseudoknot in SARS-CoV-2: Control of Ion-Mediated Pierced Lasso Topology

The programmed frameshifting stimulatory element, a promising drug target for COVID-19 treatment, involves a RNA pseudoknot (PK) structure. This RNA PK facilitates frameshifting, enabling RNA viruses to translate multiple proteins from a single mRNA, which is a key strategy for their rapid evolution. Overcoming the challenges of capturing large-scale structural changes of RNA under the influence of a dynamic counterion environment (K+ and Mg2+), the study extended the applications of a newly developed dynamic counterion condensation (DCC) model. DCC simulations reveal potential folding pathways of this RNA PK, supported by the experimental findings obtained using optical tweezers. The study elucidates the pivotal role of Mg2+ ions in crafting a lasso-like RNA topology, a novel RNA motif that governs dynamic transitions between the ring-opened and ring-closed states of the RNA. The pierced lasso component guided by Mg2+-mediated interactions orchestrates inward and outward motion fine-tuning tension on the slippery segment, a critical factor for optimizing frameshifting efficiency.

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.

Topological links in predicted protein complex structures reveal limitations of AlphaFold

AlphaFold is making great progress in protein structure prediction, not only for single-chain proteins but also for multi-chain protein complexes. When using AlphaFold-Multimer to predict protein‒protein complexes, we observed some unusual structures in which chains are looped around each other to form topologically intertwining links at the interface. Based on physical principles, such topological links should generally not exist in native protein complex structures unless covalent modifications of residues are involved. Although it is well known and has been well studied that protein structures may have topologically complex shapes such as knots and links, existing methods are hampered by the chain closure problem and show poor performance in identifying topologically linked structures in protein‒protein complexes. Therefore, we address the chain closure problem by using sliding windows from a local perspective and propose an algorithm to measure the topological-geometric features that can be used to identify topologically linked structures. An application of the method to AlphaFold-Multimer-predicted protein complex structures finds that approximately 1.72% of the predicted structures contain topological links. The method presented in this work will facilitate the computational study of protein‒protein interactions and help further improve the structural prediction of multi-chain protein complexes.

How synonymous mutations alter enzyme structure and function over long timescales

The specific activity of enzymes can be altered over long timescales in cells by synonymous mutations that alter a messenger RNA molecule's sequence but not the encoded protein's primary structure. How this happens at the molecular level is unknown. Here, we use multiscale modelling of three Escherichia coli enzymes (type III chloramphenicol acetyltransferase, D-alanine-D-alanine ligase B and dihydrofolate reductase) to understand experimentally measured changes in specific activity due to synonymous mutations. The modelling involves coarse-grained simulations of protein synthesis and post-translational behaviour, all-atom simulations to test robustness and quantum mechanics/molecular mechanics calculations to characterize enzymatic function. We show that changes in codon translation rates induced by synonymous mutations cause shifts in co-translational and post-translational folding pathways that kinetically partition molecules into subpopulations that very slowly interconvert to the native, functional state. Structurally, these states resemble the native state, with localized misfolding near the active sites of the enzymes. These long-lived states exhibit reduced catalytic activity, as shown by their increased activation energies for the reactions they catalyse.

The Zero-Order Loop in Apoazurin Modulates Folding Mechanism In Silico

Pseudomonas aeruginosa apoazurin (apo, without the copper cofactor) has a single disulfide bond between residues 3 and 26 and unfolds in a two-state reaction in vitro. The disulfide bond covalently connects the N-termini of β-strands 1 and 3; thereby, it creates a zero-order loop or a "cinch" that restricts conformational space. Covalent loops and threaded topologies are emerging as important structural elements in folded proteins and may be important for function. In order to understand the role of a zero-order loop in the folding process of a protein, here we used coarse-grained molecular dynamics (CGMD) simulations in silico to compare two variants of apoazurin: one named "loop" which contained the disulfide, and another named "open" in which the disulfide bond between residues 3 and 26 was removed. CGMD simulations were performed to probe the stability and unfolding pathway of the two apoazurin variants at different urea concentrations and temperatures. Our results show that the covalent loop plays a prominent role in the unfolding mechanism of apoazurin; its removal alters both the folding-transition state and the unfolded-state ensemble of conformations. We propose that modulation of azurin's folding landscape by the disulfide bridge may be related to both copper capturing and redox sensing.

Topological Twists in Nature

The first entangled protein was observed about 30 years ago, resulting in an increased interest for uncovering the biological functions and biophysical properties of these complex topologies. Recently, the Pierced Lasso Topology (PLT) was discovered in which a covalent bond forms an intramolecular loop, leaving one or both termini free to pierce the loop. This topology is related to knots and other entanglements. PLTs exist in many well-researched systems where the PLTs have previously been unnoticed. PLTs represents 18% of all disulfide containing proteins across all kingdoms of life. In this review, we investigate the biological implications of this specific topology in which the PLT-forming disulfide may act as a molecular switch for protein function and consequently human health.

GLN: a method to reveal unique properties of lasso type topology in proteins

Geometry and topology are the main factors that determine the functional properties of proteins. In this work, we show how to use the Gauss linking integral (GLN) in the form of a matrix diagram-for a pair of a loop and a tail-to study both the geometry and topology of proteins with closed loops e.g. lassos. We show that the GLN method is a significantly faster technique to detect entanglement in lasso proteins in comparison with other methods. Based on the GLN technique, we conduct comprehensive analysis of all proteins deposited in the PDB and compare it to the statistical properties of the polymers. We show how high and low GLN values correlate with the internal exibility of proteins, and how the GLN in the form of a matrix diagram can be used to study folding and unfolding routes. Finally, we discuss how the GLN method can be applied to study entanglement between two structures none of which are closed loops. Since this approach is much faster than other linking invariants, the next step will be evaluation of lassos in much longer molecules such as RNA or loops in a single chromosome.

Lasso Proteins: Modular Design, Cellular Synthesis, and Topological Transformation

Entangled proteins have attracted significant research interest. Herein, we report the first rationally designed lasso proteins, or protein [1]rotaxanes, by using a p53dim-entwined dimer for intramolecular entanglement and a SpyTag-SpyCatcher reaction for side-chain ring closure. The lasso structures were confirmed by proteolytic digestion, mutation, NMR spectrometry, and controlled ligation. Their dynamic properties were probed by experiments such as end-capping, proteolytic digestion, and heating/cooling. As a versatile topological intermediate, a lasso protein could be converted to a rotaxane, a heterocatenane, and a "slide-ring" network. Being entirely genetically encoded, this robust and modular lasso-protein motif is a valuable addition to the topological protein repertoire and a promising candidate for protein-based biomaterials.

The Pierced Lasso Topology Leptin has a Bolt on Dynamic Domain Composed by the Disordered Loops I and III

Leptin is an important signaling hormone, mostly known for its role in energy expenditure and satiety. Furthermore, leptin plays a major role in other proteinopathies, such as cancer, marked hyperphagia, impaired immune function, and inflammation. In spite of its biological relevance in human health, there are no NMR resonance assignments of the human protein available, obscuring high-resolution characterization of the soluble protein and/or its conformational dynamics, suggested as being important for receptor interaction and biological activity. Here, we report the nearly complete backbone resonance assignments of human leptin. Chemical shift-based secondary structure prediction confirms that in solution leptin forms a four-helix bundle including a pierced lasso topology. The conformational dynamics, determined on several timescales, show that leptin is monomeric, has a rigid four-helix scaffold, and a dynamic domain, including a transiently formed helix. The dynamic domain is anchored to the helical scaffold by a secondary hydrophobic core, pinning down the long loops of leptin to the protein body, inducing motional restriction without a well-defined secondary or tertiary hydrogen bond stabilized structure. This dynamic region is well suited for and may be involved in functional allosteric dynamics upon receptor binding.

Transient knots in intrinsically disordered proteins and neurodegeneration

We provide a brief overview of the topological features found in structured proteins and of the dynamical processes that involve knots. We then discuss the knotted states that arise in the intrinsically disordered polyglutamine and α-synuclein. We argue that the existence of the knotted conformations stalls degradation by proteases and thus enhances aggregation. This mechanism works if the length of a peptide chain exceeds a threshold, as in the Huntington disease. We also study the cavities that form within the conformations of the disordered proteins. The volume of the cavities varies in time in a way that is different than that of the radius of gyration or the end-to-end distance. In addition, we study the traffic between the conformational basins and identify patterns associated with the deep and shallow knots. The results are obtained by molecular dynamics simulations that use coarse-grained and all-atom models (with and without the explicit solvent).

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.

Directional Immobilization of Proteins on Gold Nanoparticles Is Essential for Their Biological Activity: Leptin as a Case Study

Gold nanomaterials hold great potential for biomedical applications. While this field is evolving rapidly, little attention has been paid to precise nanoparticle design and functionalization. Here, we show that when using proteins as targeting moieties, it is fundamental to immobilize them directionally to preserve their biological activity. Using full-length leptin as a case study, we have developed two alternative conjugation strategies for protein immobilization based on either a site-selective or a nonselective derivatization approach. We show that only nanoparticles with leptin immobilized site-selectively fully retain the ability to interact with the cognate leptin receptor. These results demonstrate the importance of a specified molecular design when preparing nanoparticles labeled with proteins.

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.

Supercoiling in a Protein Increases its Stability

The supercoiling motif is the most complex type of nontrivial topology found in proteins with at least one disulfide bond and, to the best of our knowledge, it has not been studied before. We show that a protein from extremophilic species with such a motif can fold; however, the supercoiling changes a smooth landscape observed in reduced conditions into a two-state folding process in the oxidative conditions, with a deep intermediate state. The protein takes advantage of the hairpinlike motif to overcome the topological barrier and thus to supercoil. We find that the depth of the supercoiling motif, i.e., the length of the threaded terminus, has a crucial impact on the folding rates of the studied protein. We show that fluctuations of the minimal surface area can be used to measure local stability, and we find that supercoiling introduces stability into the protein. We suggest that the supercoiling motif enables the studied protein to live in physically extreme conditions, which are detrimental to most life on Earth.

Searching the Optimal Folding Routes of a Complex Lasso Protein

Understanding how polypeptides can efficiently and reproducibly attain a self-entangled conformation is a compelling biophysical challenge that might shed new light on our general knowledge of protein folding. Complex lassos, namely self-entangled protein structures characterized by a covalent loop sealed by a cysteine bridge, represent an ideal test system in the framework of entangled folding. Indeed, because cysteine bridges form in oxidizing conditions, they can be used as on/off switches of the structure topology to investigate the role played by the backbone entanglement in the process. In this work, we have used molecular dynamics to simulate the folding of a complex lasso glycoprotein, granulocyte-macrophage colony-stimulating factor, modeling both reducing and oxidizing conditions. Together with a well-established Gō-like description, we have employed the elastic folder model, a coarse-grained, minimalistic representation of the polypeptide chain driven by a structure-based angular potential. The purpose of this study is to assess the kinetically optimal pathways in relation to the formation of the native topology. To this end, we have implemented an evolutionary strategy that tunes the elastic folder model potentials to maximize the folding probability within the early stages of the dynamics. The resulting protein model is capable of folding with high success rate, avoiding the kinetic traps that hamper the efficient folding in the other tested models. Employing specifically designed topological descriptors, we could observe that the selected folding routes avoid the topological bottleneck by locking the cysteine bridge after the topology is formed. These results provide valuable insights on the selection of mechanisms in self-entangled protein folding while, at the same time, the proposed methodology can complement the usage of established minimalistic models and draw useful guidelines for more detailed simulations.

Uncovering the molecular mechanisms behind disease-associated leptin variants

The pleiotropic hormone leptin has a pivotal role in regulating energy balance by inhibiting hunger and increasing energy expenditure. Homozygous mutations found in the leptin gene are associated with extreme obesity, marked hyperphagia, and impaired immune function. Although these mutations have been characterized in vivo, a detailed understanding of how they affect leptin structure and function remains elusive. In the current work, we used NMR, differential scanning calorimetry, molecular dynamics simulations, and bioinformatics calculations to characterize the effects of these mutations on leptin structure and function and binding to its cognate receptor. We found that mutations identified in patients with congenital leptin deficiency not only cause leptin misfolding or aggregation, but also cause changes in the dynamics of leptin residues on the receptor-binding interface. Therefore, we infer that mutation-induced leptin deficiency may arise from several distinct mechanisms including (i) blockade of leptin receptor interface II, (ii) decreased affinity in the second step of leptin's interaction with its receptor, (iii) leptin destabilization, and (iv) unsuccessful threading through the covalent loop, leading to leptin misfolding/aggregation. We propose that this expanded framework for understanding the mechanisms underlying leptin deficiency arising from genetic mutations may be useful in designing therapeutics for leptin-associated disorders.

Scaling behavior of knotted random polygons and self-avoiding polygons: Topological swelling with enhanced exponent

We show that the average size of self-avoiding polygons (SAPs) with a fixed knot is much larger than that of no topological constraint if the excluded volume is small and the number of segments is large. We call it topological swelling. We argue an "enhancement" of the scaling exponent for random polygons with a fixed knot. We study them systematically through SAP consisting of hard cylindrical segments with various different values of the radius of segments. Here we mean by the average size the mean-square radius of gyration. Furthermore, we show numerically that the topological balance length of a composite knot is given by the sum of those of all constituent prime knots. Here we define the topological balance length of a knot by such a number of segments that topological entropic repulsions are balanced with the knot complexity in the average size. The additivity suggests the local knot picture.

To Tie or Not to Tie? That Is the Question

In this review, we provide an overview of entangled proteins. Around 6% of protein structures deposited in the PBD are entangled, forming knots, slipknots, lassos and links. We present theoretical methods and tools that enabled discovering and classifying such structures. We discuss the advantages and disadvantages of the non-trivial topology in proteins, based on available data about folding, stability, biological properties and evolutionary conservation. We also formulate intriguing and challenging questions on the border of biophysics, bioinformatics, biology and mathematics, which arise from the discovery of an entanglement in proteins. Finally, we discuss possible applications of entangled proteins in medicine and nanotechnology, such as the chance to design super stable proteins, whose stability could be controlled by chemical potential.

Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots

We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs "topological polymers". Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology "topological polymers", for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer⁻Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it "topological swelling".

Proteins analysed as virtual knots

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the termini of the chain somehow. We introduce a new method for resolving knotting in open curves using virtual knots, which are a wider class of topological objects that do not require a classical closure and so naturally capture the topological ambiguity inherent in open curves. We describe the results of analysing proteins in the Protein Data Bank by this new scheme, recovering and extending previous knotting results, and identifying topological interest in some new cases. The statistics of virtual knots in protein chains are compared with those of open random walks and Hamiltonian subchains on cubic lattices, identifying a regime of open curves in which the virtual knotting description is likely to be important.

Pierced Lasso Topology Controls Function in Leptin

Protein engineering is a powerful tool in drug design and therapeutics, where disulphide bridges are commonly introduced to stabilize proteins. However, these bonds also introduce covalent loops, which are often neglected. These loops may entrap the protein backbone on opposite sides, leading to a "knotted" topology, forming a so-called Pierced Lasso (PL). In this elegant system, the "knot" is held together with a single disulphide bridge where part of the polypeptide chain is threaded through. The size and position of these covalent loops can be manipulated through protein design in vitro, whereas nature uses polymorphism to switch the PL topology. The PL protein leptin shows genetic modification of an N-terminal residue, adding a third cysteine to the same sequence. In an effort to understand the mechanism of threading of these diverse topologies, we designed three loop variants to mimic the polymorphic sequence. This adds elegance to the system under study, as it allows the generation of three possible covalent loops; they are the original wild-type C-terminal loop protein, the fully circularized unthreaded protein, and the N-terminal loop protein, responsible for different lasso topologies. The size of the loop changes the threading mechanism from a slipknotting to a plugging mechanism, with increasing loop size. Interestingly, the ground state of the native protein structure is largely unaffected, but biological assays show that the activity is maximized by properly controlled dynamics in the threaded state. A threaded topology with proper conformational dynamics is important for receptor interaction and activation of the signaling pathways in vivo.

Complex lasso: new entangled motifs in proteins

We identify new entangled motifs in proteins that we call complex lassos. Lassos arise in proteins with disulfide bridges (or in proteins with amide linkages), when termini of a protein backbone pierce through an auxiliary surface of minimal area, spanned on a covalent loop. We find that as much as 18% of all proteins with disulfide bridges in a non-redundant subset of PDB form complex lassos, and classify them into six distinct geometric classes, one of which resembles supercoiling known from DNA. Based on biological classification of proteins we find that lassos are much more common in viruses, plants and fungi than in other kingdoms of life. We also discuss how changes in the oxidation/reduction potential may affect the function of proteins with lassos. Lassos and associated surfaces of minimal area provide new, interesting and possessing many potential applications geometric characteristics not only of proteins, but also of other biomolecules.

Thermal Unthreading of the Lasso Peptides Astexin-2 and Astexin-3

Lasso peptides are a class of knot-like polypeptides in which the C-terminal tail of the peptide threads through a ring formed by an isopeptide bond between the N-terminal amine group and a side chain carboxylic acid. The small size (∼20 amino acids) and simple topology of lasso peptides make them a good model system for studying the unthreading of entangled polypeptides, both with experiments and atomistic simulation. Here, we present an in-depth study of the thermal unthreading behavior of two lasso peptides astexin-2 and astexin-3. Quantitative kinetics and energetics of the unthreading process were determined for variants of these peptides using a series of chromatography and mass spectrometry experiments and biased molecular dynamics (MD) simulations. In addition, we show that the Tyr15Phe variant of astexin-3 unthreads via an unprecedented "tail pulling" mechanism. MD simulations on a model ring-thread system coupled with machine learning approaches also led to the discovery of physicochemical descriptors most important for peptide unthreading.

How to fold intricately: using theory and experiments to unravel the properties of knotted proteins

Over the years, advances in experimental and computational methods have helped us to understand the role of thermodynamic, kinetic and active (chaperone-aided) effects in coordinating the folding steps required to achieving a knotted native state. Here, we review such developments by paying particular attention to the complementarity of experimental and computational studies. Key open issues that could be tackled with either or both approaches are finally pointed out.

LassoProt: server to analyze biopolymers with lassos

The LassoProt server, http://lassoprot.cent.uw.edu.pl/, enables analysis of biopolymers with entangled configurations called lassos. The server offers various ways of visualizing lasso configurations, as well as their time trajectories, with all the results and plots downloadable. Broad spectrum of applications makes LassoProt a useful tool for biologists, biophysicists, chemists, polymer physicists and mathematicians. The server and our methods have been validated on the whole PDB, and the results constitute the database of proteins with complex lassos, supported with basic biological data. This database can serve as a source of information about protein geometry and entanglement-function correlations, as a reference set in protein modeling, and for many other purposes.

Molecular knots in biology and chemistry

Knots and entanglements are ubiquitous. Beyond their aesthetic appeal, these fascinating topological entities can be either useful or cumbersome. In recent decades, the importance and prevalence of molecular knots have been increasingly recognised by scientists from different disciplines. In this review, we provide an overview on the various molecular knots found in naturally occurring biological systems (DNA, RNA and proteins), and those created by synthetic chemists. We discuss the current knowledge in these fields, including recent developments in experimental and, in some cases, computational studies which are beginning to shed light into the complex interplay between the structure, formation and properties of these topologically intricate molecules.

Engineering covalent loops in proteins can serve as an on/off switch to regulate threaded topologies

Knots in proteins are under active investigation motivating refinements of current techniques and the development of tools to better understand the knotted topology. A strong focus is to identify new knots and expand upon the current understanding of their complex topology. Previous work has shown that the knotted topology, even in the simplest case of knots, encompasses a variety of unique challenges in folding and tying a chain. To bypass many of the in vitro experimental complications involved in working with knots, it is useful to apply methodologies to a more simplified system. The pierced lasso bundles (PLB), we discovered where a single disulphide bridge holds the threaded topology together, presents a simpler system to study knots in vitro. Having a disulphide bridge as an on/off switch between the threaded/unthreaded topology is advantageous because a covalent loop allows manipulation of the knot without directly altering affecting secondary and tertiary structure. Because disulphide bridges are commonly used in protein engineering, a pierced lasso (PL) topology can be easily introduced into a protein of interest to form a knotted topology within a given secondary structure. It is also important to take into account that if formed, disulphides can inadvertently introduce an unwanted PL. This was found upon determination of the crystal structure (PDB code 2YHG) of the recently de novo designed nucleoside hydrolase. Our detailed investigations of the PL presented here will allow researchers to look at the introduction of disulphide bridges in a larger context with respect to potential geometrical consequences on the structure and functional properties of proteins.

What protein folding teaches us about biological function and molecular machines

Protein folding was the first area of molecular biology for which a systematic statistical-mechanical analysis of dynamics was developed. As a result, folding is described as a process by which a disordered protein chain diffuses across a high-dimensional energy landscape and finally reaches the folded ensemble. Folding studies have produced countless theoretical concepts that are generalizable to other biomolecular processes, such as the functional dynamics of molecular assemblies. Common themes in folding and function include the dominant role of excluded volume, that a balance between energetic roughness and geometrical effects guides dynamics, and that folding/functional landscapes are relatively smooth. Here, we discuss how insights into protein folding have been applied to investigate the functional dynamics of biomolecular assemblies.