Dodging the crisis of folding proteins with knots

Protein misfolding involving entanglements providesa structural explanation for the origin of stretched-exponential refolding kinetics

Stretched-exponential protein refolding kinetics, first observed decades ago, were attributed to a nonnative ensemble of structures with parallel, non-interconverting folding pathways. However, the structural origin of the large energy barriers preventing interconversion between these folding pathways is unknown. Here, we combine simulations with limited proteolysis (LiP) and cross-linking (XL) mass spectrometry (MS) to study the protein phosphoglycerate kinase (PGK). Simulations recapitulate its stretched-exponential folding kinetics and reveal that misfolded states involving changes of entanglement underlie this behavior: either formation of a nonnative, noncovalent lasso entanglement or failure to form a native entanglement. These misfolded states act as kinetic traps, requiring extensive unfolding to escape, which results in a distribution of free energy barriers and pathway partitioning. Using LiP-MS and XL-MS, we propose heterogeneous structural ensembles consistent with these data that represent the potential long-lived misfolded states PGK populates. This structural and energetic heterogeneity creates a hierarchy of refolding timescales, explaining stretched-exponential kinetics.

Everything AlphaFold tells us about protein knots

Recent advances in Machine Learning methods in structural biology opened up new perspectives for protein analysis. Utilizing these methods allows us to go beyond the limitations of empirical research, and take advantage of the vast amount of generated data. We use a complete set of potentially knotted protein models identified in all high-quality predictions from the AlphaFold Database to search for any common trends that describe them. We show that the vast majority of knotted proteins have 31 knot and that the presence of knots is preferred in neither Bacteria, Eukaryota, or Archaea domains. On the contrary, the percentage of knotted proteins in any given proteome is around 0.4%, regardless of the taxonomical group. We also verified that the organism's living conditions do not impact the number of knotted proteins in its proteome, as previously expected. We did not encounter an organism without a single knotted protein. What is more, we found four universally present families of knotted proteins in Bacteria, consisting of SAM synthase, and TrmD, TrmH, and RsmE methyltransferases.

Conformational and Structural Characterization of Knotted Proteins

Knotted proteins are fascinating natural biomolecules whose backbones entangle themselves in a knot. Their particular knotted configurations provide them with a wide range of topological features. However, their folding/unfolding mechanisms, stability, and function are poorly understood. In the present work, native trapped ion mobility spectrometry-mass spectrometry (TIMS-MS) was used for characterizing structural features of two model knotted proteins: a Gordian 52 knot ubiquitin C-terminal hydrolase (UCH) and a Stevedore 61 knot (α-haloacid dehalogenase, DehI). Experimental results showed structural transitions of UCH and DehI as a function of solution composition (0-50% MeOH) and temperature (T ∼20-95 °C). An increase in the protein charge states and collision cross sections (∼2750-8750 Å2 and ∼3250-15,385 Å2 for UCH and DehI, respectively) with the solution organic content (OC) and temperature suggested a three-step unfolding pathway with at least four structural transitions. Results also showed that the integrity of the UCH knot core was more resistant to thermal unfolding when compared to DehI; however, both knot cores can be disrupted with the increase in the solution OC. Additional enzymatic digestion experiments using carboxypeptidase Y combined with molecular dynamics simulations showed that the knot core was preserved between Glu20 and Glu188 and Arg89 and His304 residues for UCH and DehI, respectively, where disruption of the knot core led to structural collapse followed by unfolding events. This work highlights the potential of solution OC and temperature studies combined with native TIMS-MS for the comprehensive characterization of knotted proteins to gain a better understanding of their structural transitions.

Knot or not? Identifying unknotted proteins in knotted families with sequence-based Machine Learning model

Knotted proteins, although scarce, are crucial structural components of certain protein families, and their roles continue to be a topic of intense research. Capitalizing on the vast collection of protein structure predictions offered by AlphaFold (AF), this study computationally examines the entire UniProt database to create a robust dataset of knotted and unknotted proteins. Utilizing this dataset, we develop a machine learning (ML) model capable of accurately predicting the presence of knots in protein structures solely from their amino acid sequences. We tested the model's capabilities on 100 proteins whose structures had not yet been predicted by AF and found agreement with our local prediction in 92% cases. From the point of view of structural biology, we found that all potentially knotted proteins predicted by AF can be classified only into 17 families. This allows us to discover the presence of unknotted proteins in families with a highly conserved knot. We found only three new protein families: UCH, DUF4253, and DUF2254, that contain both knotted and unknotted proteins, and demonstrate that deletions within the knot core could potentially account for the observed unknotted (trivial) topology. Finally, we have shown that in the majority of knotted families (11 out of 15), the knotted topology is strictly conserved in functional proteins with very low sequence similarity. We have conclusively demonstrated that proteins AF predicts as unknotted are structurally accurate in their unknotted configurations. However, these proteins often represent nonfunctional fragments, lacking significant portions of the knot core (amino acid sequence).

Are there double knots in proteins? Prediction and in vitro verification based on TrmD-Tm1570 fusion from C. nitroreducens

We have been aware of the existence of knotted proteins for over 30 years-but it is hard to predict what is the most complicated knot that can be formed in proteins. Here, we show new and the most complex knotted topologies recorded to date-double trefoil knots (31 #31). We found five domain arrangements (architectures) that result in a doubly knotted structure in almost a thousand proteins. The double knot topology is found in knotted membrane proteins from the CaCA family, that function as ion transporters, in the group of carbonic anhydrases that catalyze the hydration of carbon dioxide, and in the proteins from the SPOUT superfamily that gathers 31 knotted methyltransferases with the active site-forming knot. For each family, we predict the presence of a double knot using AlphaFold and RoseTTaFold structure prediction. In the case of the TrmD-Tm1570 protein, which is a member of SPOUT superfamily, we show that it folds in vitro and is biologically active. Our results show that this protein forms a homodimeric structure and retains the ability to modify tRNA, which is the function of the single-domain TrmD protein. However, how the protein folds and is degraded remains unknown.

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.

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.

Folding and functions of knotted proteins

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

Effects of sequence-dependent non-native interactions in equilibrium and kinetic folding properties of knotted proteins

Determining the role of non-native interactions in folding dynamics, kinetics, and mechanisms is a classic problem in protein folding. More recently, this question has witnessed a renewed interest in light of the hypothesis that knotted proteins require the assistance of non-native interactions to fold efficiently. Here, we conduct extensive equilibrium and kinetic Monte Carlo simulations of a simple off-lattice C-alpha model to explore the role of non-native interactions in the thermodynamics and kinetics of three proteins embedding a trefoil knot in their native structure. We find that equilibrium knotted conformations are stabilized by non-native interactions that are non-local, and proximal to native ones, thus enhancing them. Additionally, non-native interactions increase the knotting frequency at high temperatures, and in partially folded conformations below the transition temperatures. Although non-native interactions clearly enhance the efficiency of transition from an unfolded conformation to a partially folded knotted one, they are not required to efficiently fold a knotted protein. Indeed, a native-centric interaction potential drives the most efficient folding transition, provided that the simulation temperature is well below the transition temperature of the considered model system.

How soluble misfolded proteins bypass chaperones at the molecular level

Subpopulations of soluble, misfolded proteins can bypass chaperones within cells. The extent of this phenomenon and how it happens at the molecular level are unknown. Through a meta-analysis of the experimental literature we find that in all quantitative protein refolding studies there is always a subpopulation of soluble but misfolded protein that does not fold in the presence of one or more chaperones, and can take days or longer to do so. Thus, some misfolded subpopulations commonly bypass chaperones. Using multi-scale simulation models we observe that the misfolded structures that bypass various chaperones can do so because their structures are highly native like, leading to a situation where chaperones do not distinguish between the folded and near-native-misfolded states. More broadly, these results provide a mechanism by which long-time scale changes in protein structure and function can persist in cells because some misfolded states can bypass components of the proteostasis machinery.

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.

Knotting Optimization and Folding Pathways of a Go-Model with a Deep Knot

Formation of protein knots is an intriguing offshoot of the protein folding problem. Since experimental resolution on knot formation is limited, theoretical methods currently provide the most detailed insights into the knotting process. While suitable for shallow knots, molecular dynamics simulations have faced challenges capturing the formation of deep knots in proteins such as the minimally tied trefoil α/β methyltransferase from Thermotoga maritima (MTTTM). To improve the efficiency of MTTTM knotting in Cα Go-model simulations, mutant variants of the MTTTM Go-model were investigated. Through a structure-based analysis of knotted and unknotted states, four residues (K71, R72, E75, V76) were identified to increase the knotting efficiency from 2% to 83% when their contact energies were doubled and dihedral strength around the knot loop increased. The key features of this model are (i) a C-terminal slipknot intermediate that threads the knot in a highly unstructured intermediate, (ii) the inability to knot in native-like intermediate states, and (iii) a minor population in a long-lived trap that cannot knot. Examination of residue 71-76 contacts provides a small set of potential mutants that can directly test the model's validity. In addition, the knotting optimization process developed here has broad applicability in generating knotting-efficient models of other knotted proteins.

A Note on the Effects of Linear Topology Preservation in Monte Carlo Simulations of Knotted Proteins

Monte Carlo simulations are a powerful technique and are widely used in different fields. When applied to complex molecular systems with long chains, such as those in synthetic polymers and proteins, they have the advantage of providing a fast and computationally efficient way to sample equilibrium ensembles and calculate thermodynamic and structural properties under desired conditions. Conformational Monte Carlo techniques employ a move set to perform the transitions in the simulation Markov chain. While accepted conformations must preserve the sequential bonding of the protein chain model and excluded volume among its units, the moves themselves may take the chain across itself. We call this a break in linear topology preservation. In this manuscript, we show, using simple protein models, that there is no difference in equilibrium properties calculated with a move set that preserves linear topology and one that does not. However, for complex structures, such as those of deeply knotted proteins, the preservation of linear topology provides correct equilibrium results but only after long relaxation. In any case, to analyze folding pathways, knotting mechanisms and folding kinetics, the preservation of linear topology may be an unavoidable requirement.

AlphaFold predicts the most complex protein knot and composite protein knots

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

Universal protein misfolding intermediates can bypass the proteostasis network and remain soluble and less functional

Some misfolded protein conformations can bypass proteostasis machinery and remain soluble in vivo. This is an unexpected observation, as cellular quality control mechanisms should remove misfolded proteins. Three questions, then, are: how do long-lived, soluble, misfolded proteins bypass proteostasis? How widespread are such misfolded states? And how long do they persist? We address these questions using coarse-grain molecular dynamics simulations of the synthesis, termination, and post-translational dynamics of a representative set of cytosolic E. coli proteins. We predict that half of proteins exhibit misfolded subpopulations that bypass molecular chaperones, avoid aggregation, and will not be rapidly degraded, with some misfolded states persisting for months or longer. The surface properties of these misfolded states are native-like, suggesting they will remain soluble, while self-entanglements make them long-lived kinetic traps. In terms of function, we predict that one-third of proteins can misfold into soluble less-functional states. For the heavily entangled protein glycerol-3-phosphate dehydrogenase, limited-proteolysis mass spectrometry experiments interrogating misfolded conformations of the protein are consistent with the structural changes predicted by our simulations. These results therefore provide an explanation for how proteins can misfold into soluble conformations with reduced functionality that can bypass proteostasis, and indicate, unexpectedly, this may be a wide-spread phenomenon.

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

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

Knot formation of dsDNA pushed inside a nanochannel

Recent experiments demonstrated that knots in single molecule dsDNA can be formed by compression in a nanochannel. In this manuscript, we further elucidate the underlying molecular mechanisms by carrying out a compression experiment in silico, where an equilibrated coarse-grained double-stranded DNA confined in a square channel is pushed by a piston. The probability of forming knots is a non-monotonic function of the persistence length and can be enhanced significantly by increasing the piston speed. Under compression knots are abundant and delocalized due to a backfolding mechanism from which chain-spanning loops emerge, while knots are less frequent and only weakly localized in equilibrium. Our in silico study thus provides insights into the formation, origin and control of DNA knots in nanopores.

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.

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

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

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.

A Topological Selection of Folding Pathways from Native States of Knotted Proteins

Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different folding behaviour for deeply and shallow knotted proteins. We analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins. In particular, the folding pathway of shallow knotted carbonic anhydrases involves the creation of a double-looped structure, contrary to what has been observed for other knotted trefoil proteins. We validate this with Molecular Dynamics simulations. By leveraging the geometry and local symmetries of knotted proteins’ native states, we provide the first numerical evidence of a double-loop folding mechanism in trefoil proteins.

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

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

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

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

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

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

On topology and knotty entanglement in protein folding

We investigate aspects of topology in protein folding. For this we numerically simulate the temperature driven folding and unfolding of the slipknotted archaeal virus protein AFV3-109. Due to knottiness the (un)folding is a topological process, it engages the entire backbone in a collective fashion. Accordingly we introduce a topological approach to model the process. Our simulations reveal that the (un)folding of AFV3-109 slipknot proceeds through a folding intermediate that has the topology of a trefoil knot. We observe that the final slipknot causes a slight swelling of the folded AFV3-109 structure. We disclose the relative stability of the strands and helices during both the folding and unfolding processes. We confirm results from previous studies that pointed out that it can be very demanding to simulate the formation of knotty self-entanglement, and we explain how the problems are circumvented: The slipknotted AFV3-109 protein is a very slow folder with a topologically demanding pathway, which needs to be properly accounted for in a simulation description. When we either increase the relative stiffness of bending, or when we decrease the speed of ambient cooling, the rate of slipknot formation rapidly increases.

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.

Can Soft Models Describe Polymer Knots?

Similar to macroscopic ropes and cables, long polymers create knots. We address the fundamental question whether and under which conditions it is possible to describe these intriguing objects with crude models that capture only mesoscale polymer properties. We focus on melts of long polymers which we describe by a model typical for mesoscopic simulations. A worm-like chain model defines the polymer architecture. To describe nonbonded interactions, we deliberately choose a generic "soft" repulsive potential that leads to strongly overlapping monomers and coarse local liquid structure. The soft model is parametrized to accurately reproduce mesoscopic structure and conformations of reference polymer melts described by a microscopic model. The microscopically resolved samples retain all generic features affecting polymer topology and provide, therefore, reliable reference data on knots. We compare characteristic knotting properties in mesoscopic and microscopically resolved melts for different cases of chain stiffness. We conclude that mesoscopic models can reliably describe knots in those melts, where the length scale characterizing polymer stiffness is substantially larger than the size of monomer-monomer excluded volume. In this case, simplified local liquid structure influences knotting properties only marginally. In contrast, mesoscopic models perform poorly in melts with flexible chains. We qualitatively explain our findings through a free energy model of simple knots available in the literature.

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

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

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

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.

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

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

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

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.

Order–disorder skewness in alpha-synuclein: a key mechanism to recognize membrane curvature

Currently, membrane curvature is understood as an active mechanism to control cells spatial organization and activity.

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

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

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

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

Recent Advances in Coarse-Grained Models for Biomolecules and Their Applications

Molecular dynamics simulations have emerged as a powerful tool to study biological systems at varied length and timescales. The conventional all-atom molecular dynamics simulations are being used by the wider scientific community in routine to capture the conformational dynamics and local motions. In addition, recent developments in coarse-grained models have opened the way to study the macromolecular complexes for time scales up to milliseconds. In this review, we have discussed the principle, applicability and recent development in coarse-grained models for biological systems. The potential of coarse-grained simulation has been reviewed through state-of-the-art examples of protein folding and structure prediction, self-assembly of complexes, membrane systems and carbohydrates fiber models. The multiscale simulation approaches have also been discussed in the context of their emerging role in unravelling hierarchical level information of biosystems. We conclude this review with the future scope of coarse-grained simulations as a constantly evolving tool to capture the dynamics of biosystems.

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

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

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

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

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.

Topological Disentanglement Dynamics of Torus Knots on Open Linear Polymers

We simulate and study the topological disentanglement occurring when torus knots reach the ends of a semiflexible open polymer (decay into simpler knots or unknotting). Through a rescaling procedure and the application of appropriate boundary conditions, we show that the full unknotting process can be understood in terms of point-like particles representing essential crossings, diffusing on the support [0, 1]. We address the bending and configurational free energy drives on the diffusion process, together with the scaling properties of the effective diffusion and friction coefficients. Agreement with simulations suggests universal features for these two model parameters.

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

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

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

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

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

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

Proteins' Knotty Problems

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