Complex lasso: new entangled motifs in proteins

Discovery of a trefoil knot in the RydC RNA: Challenging previous notions of RNA topology

Knots are very common in polymers, including DNA and protein molecules. Yet, no genuine knot has been identified in natural RNA molecules to date. Upon re-examining experimentally determined RNA 3D structures, we discovered a trefoil knot 31, the most basic non-trivial knot, in the RydC RNA. This knotted RNA is a member of a small family of short bacterial RNAs, whose secondary structure is characterized by an H-type pseudoknot. Molecular dynamics simulations suggest a folding pathway of the RydC RNA that starts with a native twisted loop. Based on sequence analyses and computational RNA 3D structure predictions, we postulate that this trefoil knot is a conserved feature of all RydC-related RNAs. The first discovery of a knot in a natural RNA molecule introduces a novel perspective on RNA 3D structure formation and on fundamental research on the relationship between function and spatial structure of biopolymers.

Non-covalent Lasso Entanglements in Folded Proteins: Prevalence, Functional Implications, and Evolutionary Significance

One-third of protein domains in the CATH database contain a recently discovered tertiary topological motif: non-covalent lasso entanglements, in which a segment of the protein backbone forms a loop closed by non-covalent interactions between residues and is threaded one or more times by the N- or C-terminal backbone segment. Unknown is how frequently this structural motif appears across the proteomes of organisms. And the correlation of these motifs with various classes of protein function and biological processes have not been quantified. Here, using a combination of protein crystal structures, AlphaFold2 predictions, and Gene Ontology terms we show that in E. coli, S. cerevisiae and H. sapiens that 71%, 52% and 49% of globular proteins contain one-or-more non-covalent lasso entanglements in their native fold, and that some of these are highly complex with multiple threading events. Further, proteins containing these tertiary motifs are consistently enriched in certain functions and biological processes across these organisms and depleted in others, strongly indicating an influence of evolutionary selection pressures acting positively and negatively on the distribution of these motifs. Together, these results demonstrate that non-covalent lasso entanglements are widespread and indicate they may be extensively utilized for protein function and subcellular processes, thus impacting phenotype.

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.

Lasso Peptides: Exploring the Folding Landscape of Nature's Smallest Interlocked Motifs

Lasso peptides make up a class of natural products characterized by a threaded structure. Given their small size and stability, chemical synthesis would offer tremendous potential for the development of novel therapeutics. However, the accessibility of the pre-folded lasso architecture has limited this advance. To better understand the folding process de novo, simulations are used herein to characterize the folding propensity of microcin J25 (MccJ25), a lasso peptide known for its antimicrobial properties. New algorithms are developed to unambiguously distinguish threaded from nonthreaded precursors and determine handedness, a key feature in natural lasso peptides. We find that MccJ25 indeed forms right-handed pre-lassos, in contrast to past predictions but consistent with all natural lasso peptides. Additionally, the native pre-lasso structure is shown to be metastable prior to ring formation but to readily transition to entropically favored unfolded and nonthreaded structures, suggesting that de novo lasso folding is rare. However, by altering the ring forming residues and appending thiol and thioester functionalities, we are able to increase the stability of pre-lasso conformations. Furthermore, conditions leading to protonation of a histidine imidazole side chain further stabilize the modified pre-lasso ensemble. This work highlights the use of computational methods to characterize lasso folding and demonstrates that de novo access to lasso structures can be facilitated by optimizing sequence, unnatural modifications, and reaction conditions like pH.

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.

Distinguishing Protein Chemical Topologies Using Supercharging Ion Mobility Spectrometry-Mass Spectrometry

A technique combining ion mobility spectrometry-mass spectrometry (IMS-MS) and supercharging electrospray ionization (ESI) has been demonstrated to differentiate protein chemical topology effectively. Incorporating as many charges as possible into proteins via supercharging ESI allows the protein chains to be largely unfolded and stretched, revealing their hidden chemical topology. Different chemical topologies result in differing geometrical sizes of the unfolded proteins due to constraints in torsional rotations in cyclic domains. By introducing new topological indices, such as the chain-length-normalized collision cross-section (CCS) and the maximum charge state (zM ) in the extensively unfolded state, we were able to successfully differentiate various protein chemical topologies, including linear chains, ring-containing topologies (lasso, tadpole, multicyclics, etc.), and mechanically interlocked rings, like catenanes.

Conservation of knotted and slipknotted topology in transmembrane transporters

The existence of nontrivial topology is well accepted in globular proteins but not in membrane proteins. Our comprehensive topological analysis of the Protein Data Bank structures reveals 18 families of transmembrane proteins with nontrivial topology, showing that they constitute a significant number of membrane proteins. Moreover, we found that they comprise one of the largest groups of secondary active transporters. We classified them based on their knotted fingerprint into four groups: three slipknotted and one knotted. Unexpectedly, we found that the same protein can possess two distinct slipknot motifs that correspond to its outward- and inward-open conformational state. Based on the analysis of structures and knotted fingerprints, we show that slipknot topology is directly involved in the conformational transition and substrate transfer. Therefore, entanglement can be used to classify proteins and to find their structure-function relationship. Furthermore, based on the topological analysis of the transmembrane protein structures predicted by AlphaFold, we identified new potentially slipknotted protein families.

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.

Folding kinetics of an entangled protein

The possibility of the protein backbone adopting lasso-like entangled motifs has attracted increasing attention. After discovering the surprising abundance of natively entangled protein domain structures, it was shown that misfolded entangled subpopulations might become thermosensitive or escape the homeostasis network just after translation. To investigate the role of entanglement in shaping folding kinetics, we introduce a novel indicator and analyze simulations of a coarse-grained, structure-based model for two small single-domain proteins. The model recapitulates the well-known two-state folding mechanism of a non-entangled SH3 domain. However, despite its small size, a natively entangled antifreeze RD1 protein displays a rich refolding behavior, populating two distinct kinetic intermediates: a short-lived, entangled, near-unfolded state and a longer-lived, non-entangled, near-native state. The former directs refolding along a fast pathway, whereas the latter is a kinetic trap, consistently with known experimental evidence of two different characteristic times. Upon trapping, the natively entangled loop folds without being threaded by the N-terminal residues. After trapping, the native entangled structure emerges by either backtracking to the unfolded state or threading through the already formed but not yet entangled loop. Along the fast pathway, trapping does not occur because the native contacts at the closure of the lasso-like loop fold after those involved in the N-terminal thread, confirming previous predictions. Despite this, entanglement may appear already in unfolded configurations. Remarkably, a longer-lived, near-native intermediate, with non-native entanglement properties, recalls what was observed in cotranslational folding.

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.

A tile model of circuit topology for self-entangled biopolymers

Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and connected sums of these knots. This includes all proteins knots that have thus far been identified.

Entangled Motifs in Membrane Protein Structures

Entangled motifs are found in one-third of protein domain structures, a reference set that contains mostly globular proteins. Their properties suggest a connection with co-translational folding. Here, we wish to investigate the presence and properties of entangled motifs in membrane protein structures. From existing databases, we build a non-redundant data set of membrane protein domains, annotated with the monotopic/transmembrane and peripheral/integral labels. We evaluate the presence of entangled motifs using the Gaussian entanglement indicator. We find that entangled motifs appear in one-fifth of transmembrane and one-fourth of monotopic proteins. Surprisingly, the main features of the distribution of the values of the entanglement indicator are similar to the reference case of general proteins. The distribution is conserved across different organisms. Differences with respect to the reference set emerge when considering the chirality of entangled motifs. Although the same chirality bias is found for single-winding motifs in both membrane and reference proteins, the bias is reversed, strikingly, for double-winding motifs only in the reference set. We speculate that these observations can be rationalized in terms of the constraints exerted on the nascent chain by the co-translational bio-genesis machinery, which is different for membrane and globular proteins.

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 Local Topological Free Energy of the SARS-CoV-2 Spike Protein

The novel coronavirus SARS-CoV-2 infects human cells using a mechanism that involves binding and structural rearrangement of its Spike protein. Understanding protein rearrangement and identifying specific amino acids where mutations affect protein rearrangement has attracted much attention for drug development. In this manuscript, we use a mathematical method to characterize the local topology/geometry of the SARS-CoV-2 Spike protein backbone. Our results show that local conformational changes in the FP, HR1, and CH domains are associated with global conformational changes in the RBD domain. The SARS-CoV-2 variants analyzed in this manuscript (alpha, beta, gamma, delta Mink, G614, N501) show differences in the local conformations of the FP, HR1, and CH domains as well. Finally, most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function. Namely, the residues 484, 570, 614, 796, and 969, which are present in variants of concern and are targeted as important in protein function, are predicted as such from our model.

The protein folding rate and the geometry and topology of the native state

Proteins fold in 3-dimensional conformations which are important for their function. Characterizing the global conformation of proteins rigorously and separating secondary structure effects from topological effects is a challenge. New developments in applied knot theory allow to characterize the topological characteristics of proteins (knotted or not). By analyzing a small set of two-state and multi-state proteins with no knots or slipknots, our results show that 95.4% of the analyzed proteins have non-trivial topological characteristics, as reflected by the second Vassiliev measure, and that the logarithm of the experimental protein folding rate depends on both the local geometry and the topology of the protein’s native state.

Covalent [2]Catenane and [2]Rotaxane Synthesis via a δ-Amino Acid Template

Despite the advances in the synthesis of mechanically interlocked molecules, a generally applicable approach to interlocked natural products, such as lasso peptides, is yet to be formulated. While amino acid sequences have been introduced into several rotaxanes, the key structural components have always been dictated by the method used for supramolecular preorganization. In this work, we report the use of an ester-functionalized, aromatic δ-amino acid as the central covalent templating unit in the synthesis of both a [2]catenane and a [2]rotaxane from the same multimacrocyclic intermediate. This represents a key step toward future synthetic peptide-based interlocked products.

Lasso Proteins-Unifying Cysteine Knots and Miniproteins

Complex lasso proteins are a recently identified class of biological compounds that are present in considerable fraction of proteins with disulfide bridges. In this work, we look at complex lasso proteins as a generalization of well-known cysteine knots and miniproteins (lasso peptides). In particular, we show that complex lasso proteins with the same crucial topological features-cysteine knots and lasso peptides-are antimicrobial proteins, which suggests that they act as a molecular plug. Based on an analysis of the stability of the lasso piercing residue, we also introduce a method to determine which lasso motif is potentially functional. Using this method, we show that the lasso motif in antimicrobial proteins, as well in that in cytokines, is functionally relevant. We also study the evolution of lasso motifs, their conservation, and the usefulness of the lasso fingerprint, which extracts all topologically non-triviality concerning covalent loops. The work is completed by the presentation of extensive statistics on complex lasso proteins to analyze, in particular, the strange propensity for "negative" piercings. We also identify 21 previously unknown complex lasso proteins with an ester and a thioester bridge.

Higher Order Protein Catenation Leads to an Artificial Antibody with Enhanced Affinity and In Vivo Stability

The chemical topology is a unique dimension for protein engineering, yet the topological diversity and architectural complexity of proteins remain largely untapped. Herein, we report the biosynthesis of complex topological proteins using a rationally engineered, cross-entwining peptide heterodimer motif derived from p53dim (an entangled homodimeric mutant of the tetramerization domain of the tumor suppressor protein p53). The incorporation of an electrostatic interaction at specific sites converts the p53dim homodimer motif into a pair of heterodimer motifs with high specificity for directing chain entanglement upon folding. Its combination with split-intein-mediated ligation and/or SpyTag/SpyCatcher chemistry facilitates the programmed synthesis of protein heterocatenane or [n]catenanes in cells, leading to a general and modular approach to complex protein catenanes containing various proteins of interest. Concatenation enhances not only the target protein's affinity but also the in vivo stability as shown by its prolonged circulation time in blood. As a proof of concept, artificial antibodies have been developed by embedding a human epidermal growth factor receptor 2-specific affibody onto the [n]catenane scaffolds and shown to exhibit a higher affinity and a better pharmacokinetic profile than the wild-type affibody. These results suggest that topology engineering holds great promise in the development of therapeutic proteins.

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.

Protein self-entanglement modulates successful folding to the native state: A multi-scale modeling study

The computer-aided investigation of protein folding has greatly benefited from coarse-grained models, that is, simplified representations at a resolution level lower than atomistic, providing access to qualitative and quantitative details of the folding process that would be hardly attainable, via all-atom descriptions, for medium to long molecules. Nonetheless, the effectiveness of low-resolution models is itself hampered by the presence, in a small but significant number of proteins, of nontrivial topological self-entanglements. Features such as native state knots or slipknots introduce conformational bottlenecks, affecting the probability to fold into the correct conformation; this limitation is particularly severe in the context of coarse-grained models. In this work, we tackle the relationship between folding probability, protein folding pathway, and protein topology in a set of proteins with a nontrivial degree of topological complexity. To avoid or mitigate the risk of incurring in kinetic traps, we make use of the elastic folder model, a coarse-grained model based on angular potentials optimized toward successful folding via a genetic procedure. This light-weight representation allows us to estimate in silico folding probabilities, which we find to anti-correlate with a measure of topological complexity as well as to correlate remarkably well with experimental measurements of the folding rate. These results strengthen the hypothesis that the topological complexity of the native state decreases the folding probability and that the force-field optimization mimics the evolutionary process these proteins have undergone to avoid kinetic traps.

The local topological free energy of proteins

Protein folding, the process by which proteins attain a 3-dimensional conformation necessary for their function, remains an important unsolved problem in biology. A major gap in our understanding is how local properties of proteins relate to their global properties. In this manuscript, we use the Writhe and Torsion to introduce a new local topological/geometrical free energy that can be associated to 4 consecutive amino acids along the protein backbone. By analyzing a culled protein dataset from the PDB, our results show that high local topological free energy conformations are independent of sequence and may be involved in the rate limiting step in protein folding. By analyzing a set of 2-state single domain proteins, we find that the total local topological free energy of these proteins correlates with the experimentally observed folding rates reported in Plaxco et al. (2000).

Topoly: Python package to analyze topology of polymers

The increasing role of topology in (bio)physical properties of matter creates a need for an efficient method of detecting the topology of a (bio)polymer. However, the existing tools allow one to classify only the simplest knots and cannot be used in automated sample analysis. To answer this need, we created the Topoly Python package. This package enables the distinguishing of knots, slipknots, links and spatial graphs through the calculation of different topological polynomial invariants. It also enables one to create the minimal spanning surface on a given loop, e.g. to detect a lasso motif or to generate random closed polymers. It is capable of reading various file formats, including PDB. The extensive documentation along with test cases and the simplicity of the Python programming language make it a very simple to use yet powerful tool, suitable even for inexperienced users. Topoly can be obtained from https://topoly.cent.uw.edu.pl.

The CFTR P67L variant reveals a key role for N-terminal lasso helices in channel folding, maturation, and pharmacologic rescue

Patients with cystic fibrosis (CF) harboring the P67L variant in the cystic fibrosis transmembrane conductance regulator (CFTR) often exhibit a typical CF phenotype, including severe respiratory compromise. This rare mutation (reported in <300 patients worldwide) responds robustly to CFTR correctors, such as lumacaftor and tezacaftor, with rescue in model systems that far exceed what can be achieved for the archetypical CFTR mutant F508del. However, the specific molecular consequences of the P67L mutation are poorly characterized. In this study, we conducted biochemical measurements following low-temperature growth and/or intragenic suppression, which suggest a mechanism underlying P67L that (1) shares key pathogenic features with F508del, including off-pathway (non-native) folding intermediates, (2) is linked to folding stability of nucleotide-binding domains 1 and 2, and (3) demonstrates pharmacologic rescue that requires domains in the carboxyl half of the protein. We also investigated the "lasso" helices 1 and 2, which occur immediately upstream of P67. Based on limited proteolysis, pulse chase, and molecular dynamics analysis of full-length CFTR and a series of deletion constructs, we argue that P67L and other maturational processing (class 2) defects impair the integrity of the lasso motif and confer misfolding of downstream domains. Thus, amino-terminal missense variants elicit a conformational change throughout CFTR that abrogates maturation while providing a robust substrate for pharmacologic repair.

Quantifying steric hindrance and topological obstruction to protein structure superposition

BACKGROUND

In computational structural biology, structure comparison is fundamental for our understanding of proteins. Structure comparison is, e.g., algorithmically the starting point for computational studies of structural evolution and it guides our efforts to predict protein structures from their amino acid sequences. Most methods for structural alignment of protein structures optimize the distances between aligned and superimposed residue pairs, i.e., the distances traveled by the aligned and superimposed residues during linear interpolation. Considering such a linear interpolation, these methods do not differentiate if there is room for the interpolation, if it causes steric clashes, or more severely, if it changes the topology of the compared protein backbone curves.

RESULTS

To distinguish such cases, we analyze the linear interpolation between two aligned and superimposed backbones. We quantify the amount of steric clashes and find all self-intersections in a linear backbone interpolation. To determine if the self-intersections alter the protein's backbone curve significantly or not, we present a path-finding algorithm that checks if there exists a self-avoiding path in a neighborhood of the linear interpolation. A new path is constructed by altering the linear interpolation using a novel interpretation of Reidemeister moves from knot theory working on three-dimensional curves rather than on knot diagrams. Either the algorithm finds a self-avoiding path or it returns a smallest set of essential self-intersections. Each of these indicates a significant difference between the folds of the aligned protein structures. As expected, we find at least one essential self-intersection separating most unknotted structures from a knotted structure, and we find even larger motions in proteins connected by obstruction free linear interpolations. We also find examples of homologous proteins that are differently threaded, and we find many distinct folds connected by longer but simple deformations. TM-align is one of the most restrictive alignment programs. With standard parameters, it only aligns residues superimposed within 5 Ångström distance. We find 42165 topological obstructions between aligned parts in 142068 TM-alignments. Thus, this restrictive alignment procedure still allows topological dissimilarity of the aligned parts.

CONCLUSIONS

Based on the data we conclude that our program ProteinAlignmentObstruction provides significant additional information to alignment scores based solely on distances between aligned and superimposed residue pairs.

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.

Cellular Synthesis and X-ray Crystal Structure of a Designed Protein Heterocatenane

Herein, we report the biosynthesis of protein heterocatenanes using a programmed sequence of multiple post-translational processing events including intramolecular chain entanglement, in situ backbone cleavage, and spontaneous cyclization. The approach is general, autonomous, and can obviate the need for any additional enzymes. The catenane topology was convincingly proven using a combination of SDS-PAGE, LC-MS, size exclusion chromatography, controlled proteolytic digestion, and protein crystallography. The X-ray crystal structure clearly shows two mechanically interlocked protein rings with intact folded domains. It opens new avenues in the nascent field of protein-topology engineering.

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.

GISA: using Gauss Integrals to identify rare conformations in protein structures

The native structure of a protein is important for its function, and therefore methods for exploring protein structures have attracted much research. However, rather few methods are sensitive to topologic-geometric features, the examples being knots, slipknots, lassos, links, and pokes, and with each method aimed only for a specific set of such configurations. We here propose a general method which transforms a structure into a ”fingerprint of topological-geometric values” consisting in a series of real-valued descriptors from mathematical Knot Theory. The extent to which a structure contains unusual configurations can then be judged from this fingerprint. The method is not confined to a particular pre-defined topology or geometry (like a knot or a poke), and so, unlike existing methods, it is general. To achieve this our new algorithm, GISA, as a key novelty produces the descriptors, so called Gauss integrals, not only for the full chains of a protein but for all its sub-chains. This allows fingerprinting on any scale from local to global. The Gauss integrals are known to be effective descriptors of global protein folds. Applying GISA to sets of several thousand high resolution structures, we first show how the most basic Gauss integral, the writhe, enables swift identification of pre-defined geometries such as pokes and links. We then apply GISA with no restrictions on geometry, to show how it allows identifying rare conformations by finding rare invariant values only. In this unrestricted search, pokes and links are still found, but also knotted conformations, as well as more highly entangled configurations not previously described. Thus, an application of the basic scan method in GISA’s tool-box revealed 10 known cases of knots as the top positive writhe cases, while placing at the top of the negative writhe 14 cases in cis-trans isomerases sharing a spatial motif of little secondary structure content, which possibly has gone unnoticed. Possible general applications of GISA are fold classification and structural alignment based on local Gauss integrals. Others include finding errors in protein models and identifying unusual conformations that might be important for protein folding and function. By its broad potential, we believe that GISA will be of general benefit to the structural bioinformatics community. GISA is coded in C and comes as a command line tool. Source and compiled code for GISA plus read-me and examples are publicly available at GitHub (https://github.com).

Threading-Induced Dynamical Transition in Tadpole-Shaped Polymers

The relationship between polymer topology and bulk rheology remains a key question in soft matter physics. Architecture-specific constraints (or threadings) are thought to control the dynamics of ring polymers in ring-linear blends, which thus affects the viscosity to range between that of the pure rings and a value larger, but still comparable to, that of the pure linear melt. Here we consider qualitatively different systems of linear and ring polymers, fused together in "chimeric" architectures. The simplest example of this family is a "tadpole"-shaped polymer, a single ring fused to the end of a single linear chain. We show that polymers with this architecture display a threading-induced dynamical transition that substantially slows chain relaxation. Our findings shed light on how threadings control dynamics and may inform design principles for chimeric polymers with topologically tunable bulk rheological properties.

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.

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.

Folding Rate Optimization Promotes Frustrated Interactions in Entangled Protein Structures

Many native structures of proteins accomodate complex topological motifs such as knots, lassos, and other geometrical entanglements. How proteins can fold quickly even in the presence of such topological obstacles is a debated question in structural biology. Recently, the hypothesis that energetic frustration might be a mechanism to avoid topological frustration has been put forward based on the empirical observation that loops involved in entanglements are stabilized by weak interactions between amino-acids at their extrema. To verify this idea, we use a toy lattice model for the folding of proteins into two almost identical structures, one entangled and one not. As expected, the folding time is longer when random sequences folds into the entangled structure. This holds also under an evolutionary pressure simulated by optimizing the folding time. It turns out that optmized protein sequences in the entangled structure are in fact characterized by frustrated interactions at the closures of entangled loops. This phenomenon is much less enhanced in the control case where the entanglement is not present. Our findings, which are in agreement with experimental observations, corroborate the idea that an evolutionary pressure shapes the folding funnel to avoid topological and kinetic traps.

Dynamics, nanomechanics and signal transduction in reelin repeats

Reelin is a large glycoprotein controlling brain development and cell adhesion. It regulates the positioning of neurons, as well as neurotransmission and memory formation. Perturbations in reelin signaling are linked to psychiatric disorders. Reelin participates in signal transduction by binding to the lipoprotein receptors VLDLR and ApoER2 through its central region. This part is rich in repeating BNR-EGF-BNR modules. We used standard molecular dynamics, steered molecular dynamics, and perturbation response scanning computational methods to characterize unique dynamical properties of reelin modules involved in signaling. Each module has specific sensors and effectors arranged in a similar topology. In the modules studied, disulfide bridges play a protective role, probably making both selective binding and protease activity of reelin possible. Results of single reelin molecule stretching by atomic force microscopy provide the first data on the mechanical stability of individual reelin domains. The forces required for partial unfolding of the modules studied are below 60 pN.

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.

Defining and detecting links in chromosomes

Sophisticated methods for mapping chromatin contacts enable to generate data of the genome structure that provide deep insights into the formation of chromatin interactions within cell nuclei. Due to the recent progress in this field, three-dimensional genomic structures of individual haploid mouse embryonic stem cells have been determined. Here, we analyze these data (8 cells) and determine comprehensive landscape of entanglements between interphase chromosomes. We find a significant number of stable links formed by chromosome pairs. Some links are even conserved between cells. Moreover, examples of stable multiple links, with at least three chromosomes engaged, are also identified. Types of links and their location along chromosomes are determined based on computations of HOMFLY-PT polynomials and Gauss Linking Numbers. Furthermore, stability of links is studied between different models, cells, and based on relaxation simulations of the genomic structure in a simplified structure-based representation. Identified links suggest that small fraction of chromosomes are entangled not only locally. How topoisomerases engineer such configurations remains an open question. Furthermore, presented methods can be used as a quantitative assessment - descriptor - to distinguish the quality of modeled data.

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.

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.

Statistical Properties of Lasso-Shape Polymers and Their Implications for Complex Lasso Proteins Function

The shape and properties of closed loops depend on various topological factors. One of them is loop-threading, which is present in complex lasso proteins. In this work, we analyze the probability of loop-threading by the tail and its influence on the shape of the loop measured by the radius of gyration, distention, asphericity, and prolateness. In particular, we show that the probability of a trivial lasso for phantom polymer is non-zero even for an infinite structure, as well as that the threading flattens the loop by restricting its motion in one dimension. These results are further used to show that there are fewer non-trivial protein lassos than expected and select potentially functional complex lasso proteins.

Protein Knotting by Active Threading of Nascent Polypeptide Chain Exiting from the Ribosome Exit Channel

The mechanism of folding of deeply knotted proteins into their native structure is still not understood. Current thinking about protein folding is dominated by the Anfinsen dogma, stating that the structure of the folded proteins is uniquely dictated by the amino acid sequence of a given protein and that the folding is driven uniquely by the energy gained from interactions between amino acids that contact each other in the native structure of the protein. The role of ribosomes in protein folding was only seen as permitting the folding to progress from the N-terminal part of nascent protein chains. We propose here that ribosomes can participate actively in the folding of knotted proteins by actively threading nascent chains emerging from the ribosome exit channels through loops formed by a synthesized earlier portion of the same protein. Our simulations of folding of deeply knotted protein Tp0624 positively verify the proposed ribosome-driven active threading mechanism leading to the formation of deeply knotted proteins.

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.