KnotProt 2.0: a database of proteins with knots and other entangled structures

Computational discovery and systematic analysis of protein entangling motifs in nature: from algorithm to database

Nontrivial protein topology has the potential to revolutionize protein engineering by enabling the manipulation of proteins' stability and dynamics. However, the rarity of topological proteins in nature poses a challenge for their design, synthesis and application, primarily due to the limited number of available entangling motifs as synthetic templates. Discovering these motifs is particularly difficult, as entanglement is a subtle structural feature that is not readily discernible from protein sequences. In this study, we developed a streamlined workflow enabling efficient and accurate identification of structurally reliable and applicable entangling motifs from protein sequences. Through this workflow, we automatically curated a database of 1115 entangling protein motifs from over 100 thousand sequences in the UniProt Knowledgebase. In our database, 73.3% of C2 entangling motifs and 80.1% of C3 entangling motifs exhibited low structural similarity to known protein structures. The entangled structures in the database were categorized into different groups and their functional and biological significance were analyzed. The results were summarized in an online database accessible through a user-friendly web platform, providing researchers with an expanded toolbox of entangling motifs. This resource is poised to significantly advance the field of protein topology engineering and inspire new research directions in protein design and application.

Universe of Lasso Proteins: Exploring the limit of entanglement of protein predicted by AlphaFold

Knots and lasso topology represent a class of natural motifs found in proteins that are characterized by a threaded structure. Proteins with a lasso motif represent a macroscopic version of the peptide lasso, which is known for its high stability and offers tremendous potential for the development of novel therapeutics. Here, based on AlphaFold, we have shown the limit of topological complexity of naturally occurring protein structures with cysteine bridges. Based on 176 million high confidence (pLDDT > 70) AlphaFold-predicted protein models and a detailed analysis of the conservation of the motif in a family, we found four new lasso motifs, including L4 and LS4LS3 topologies, and the first examples of knotted lasso proteins: L1K31 and L3#K31. We show that in the case of natural proteins, there are no lassos with 5 threadings, but there exist some with 6. Families possessing proteins with more than 6 threadings did not exceed the conservation threshold of 10%. Moreover, we propose a probable folding mechanism for the LS4LS3 lasso motif, enhancing our view on protein folding and stability. This work expands the topological space of lasso type motifs in proteins but also suggests that more complex structures could be unfavorable for proteins.

Effect of a Knot on the Thermal Stability of Protein MJ0366: Insights into Molecular Dynamics and Monte Carlo Simulations

Protein MJ0366 is a hypothetical protein from Methanocaldococcus jannaschii that has a rare and complex knot in its structure. The knot is a right-handed trefoil knot that involves about half of the protein's residues. In this work, we investigate the thermal stability of protein MJ0366 using numerical simulations based on molecular dynamics and Monte Carlo methods. We compare the results with those of a similar unknotted protein and analyze the effects of the knot on the folding and unfolding processes. We show that the knot in protein MJ0366 increases its thermal stability by creating a topological barrier that prevents the protein from unfolding at high temperatures.

Geometry based prediction of tau protein sites and motifs associated with misfolding and aggregation

Recent studies of tau proteins point to specific sites or motifs along the protein related to its misfolding and aggregation propensity, which is associated with neurodegenerative diseases of structure-dependent pathology. In this manuscript we employ topology and geometry to analyze the local structure of tau proteins obtained from the Protein Data Bank. Our results show that mathematical topology/geometry of cryo-EM structures alone identify the PGGG motifs, and the PHF6(*) motifs as sites of interest and reveal a geometrical hierarchy of the PGGG motifs that differs for 3R+4R and 4R tauopathies. By employing the Local Topological Free Energy (LTE), we find that progressive supranuclear palsy (PSP) and globular glial tauopathy (GGT) have the highest LTE values around residues 302-305, which are inside the jR2R3 peptide and in the vicinity of the 301 site, experimentally associated with aggregation. By extending the LTE definition to estimate a global topological free energy, we find that the jR2R3 peptide of PSP and GGT, has in fact the lowest global topological free energy among other tauopathies. These results point to a possible correlation between the global topological free energy of parts of the protein and the LTE of specific sites.

Deep Learning-Assisted Discovery of Protein Entangling Motifs

Natural topological proteins exhibit unique properties including enhanced stability, controlled quaternary structures, and dynamic switching properties, highlighting topology as a unique dimension in protein engineering. Although artificial design and synthesis of topological proteins have achieved certain success, their diversity and complexity remain rather limited due to the scarcity of available entangling motifs essential for the construction of nontrivial protein topologies. In this work, we developed a deep-learning model to predict the entanglement features of a homodimer based solely on its amino acid sequence via the Gauss linking number matrices. The model achieved a search speed that was dozens of times faster than AlphaFold-Multimer, while maintaining comparable mean squared error. It was used to screen for entangling motifs from the genome of a hyperthermophilic archaeon. We demonstrated the effectiveness of our model by successful wet-lab synthesis of protein catenanes using two candidate entangling motifs. These findings show the great potential of our model for advancing the design and synthesis of novel topological proteins.

Folding of a tandemly knotted protein: Evidence that a polypeptide chain can get out of deep kinetic traps

It is hard to imagine how proteins can thread and form knots in their polypeptide chains, but they do. These topologically complex structures have challenged the traditional protein folding views of simple funnel-shaped energy landscapes. Previous experimental studies on the folding mechanisms of deeply knotted proteins with a single trefoil knot have yielded evidence that this topology has a more complicated folding landscape than other simpler proteins. However, to date, there have been no attempts to study the folding of any protein in which multiple threading events are needed to create more than one knot within a single polypeptide chain. Here, we report the construction and characterization of an artificial tandemly knotted protein. We find compelling evidence that both domains of the protein form trefoil knots with similar structures and stabilities to the parent single trefoil-knotted protein. In addition, we show that this tandemly knotted protein has a complex folding pathway in which there are additional very slow folding phases that we propose correspond to the formation of the second knot within the system. We also find evidence that during folding this protein gets transiently trapped in deep kinetic traps, however, the majority of protein chains (>90%) manage to partially unfold and acquire the native tandem-knot topology. This work highlights the fact that Nature can tolerate more complex protein topologies than we thought, and despite considerable misfolding during folding, protein chains can find their way to the native state even in the absence of molecular chaperones.

The Complex Energy Landscape of miRFP709, a 41-Knotted Protein, Results in Its Irreversible Denaturation

Knotted proteins have a unique topological feature with an open knot, and the physiological significance of these knots remains elusive. In addition, these proteins challenge our understanding of the protein folding process, and whether they retain their native state during unfolding/refolding cycles like other proteins is debated. Most folding studies on knotted proteins have been performed on 31 and 52 knots, monitoring the tryptophan fluorescence. In this study, we probe the unfolding/refolding of a 41-knotted protein, miRFP709, which can be monitored through near-infrared fluorescence in addition to the intrinsic tryptophan emission. miRFP709, upon chemical unfolding and refolding, folds back to a compact, non-native, stable structure that loses its ability to bind to the biliverdin ligand and fluoresce. The refolded protein retains its secondary structure but behaves like a molten-globule state with an exposed hydrophobic surface. The complex folding landscape of these proteins results in hysteresis between the folding and refolding curves. We propose that upon refolding, either an altered knot or an unknotted structure prevents the formation of the native knotted structure.

Knot data analysis using multiscale Gauss link integral

In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein-ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area-knot deep learning-in data science.

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.

Theta-curves in proteins

We study and characterize the topology of connectivity circuits observed in natively folded protein structures whose coordinates are deposited in the Protein Data Bank (PDB). Polypeptide chains of some proteins naturally fold into unique knotted configurations. Another kind of nontrivial topology of polypeptide chains is observed when, in addition to covalent bonds connecting consecutive amino acids in polypeptide chains, one also considers disulfide and ionic bonds between non-consecutive amino acids. Bonds between non-consecutive amino acids introduce bifurcation points into connectivity circuits defined by bonds between consecutive and nonconsecutive amino acids in analyzed proteins. Circuits with bifurcation points can form θ-curves with various topologies. We catalog here the observed topologies of θ-curves passing through bridges between consecutive and non-consecutive amino acids in studied proteins.

Mechanically Interlocked Water-Soluble Pd6 Host for the Selective Separation of Coal Tar-Based Planar Aromatic Molecules

Research on the synthesis of catenated cages has been a growing field of interest in the past few years. While multiple types of catenated cages with different structures have been synthesized, the application of such systems has been much less explored. Specifically, the use of catenated cages in the separation of industrially relevant molecules that are present in coal tar has not been explored before. Herein, we demonstrate the use of a newly synthesized interlocked cage 1 [C184H240N76O48Pd6] (M6L4), formed through the self-assembly of ligand L.HNO3 (tris(4-(1H-imidazole-1-yl)benzylidene)hydrazine-1-carbohydrazonhydrazide) with acceptor cis-[(tmchda)Pd(NO3)2] [tmchda = ±N,N,N',N'-tetramethylcyclohexane-1,2-diamine] (M). The interlocked cage 1 was able to separate the isomers (anthracene and phenanthrene) using a simple solvent extraction technique. Using the same technique, the much more difficult separation of structurally and physiochemically similar compounds acenaphthene and acenaphthylene was performed for the first time with 1 as the host. Other noninterlocked hexanuclear Pd6 cages having a wider cavity proved inefficient for such separation, demonstrating the uniqueness of the interlocked cage 1 for such challenging separation.

Exploring the electronic structure of knotted proteins: the case of two ornithine transcarbamylase family

CONTEXT

Geometrical knots are rare structural arrangements in proteins in which the polypeptide chain ties itself into a knot, which is very intriguing due to the uncertainty of their impact on the protein properties. Presently, classical molecular dynamics is the most employed technique in the few studies found on this topic, so any information on how the presence of knots affects the reactivity and electronic properties of proteins is even scarcer. Using the electronic structure methods and quantum chemical descriptors analysis, we found that the same amino-acid residues in the knot core have statistically larger values for the unknotted protein, for both hard-hard and soft-soft interaction descriptors. In addition, we present a computationally feasible protocol, where we show it is possible to separate the contribution of the geometrical knot to the reactivity and other electronic structure properties.

METHODS

In order to investigate these systems, we used PRIMoRDiA, a new software developed by our research group, to explore the electronic structure of biological macromolecules. We evaluated several local quantum chemical descriptors to unveil relevant patterns potentially originating from the presence of the geometrical knot in two proteins, belonging to the ornithine transcarbamylase family. We compared several sampled structures from these two enzymes that are highly similar in both tertiary structure and function, but one of them has a knot whereas the other does not. The sampling was carried out through molecular dynamics simulations using ff14SB force field along 50 ns, and the semiempirical convergence was performed with PM7 Hamiltonian.

AlphaKnot 2.0: a web server for the visualization of proteins' knotting and a database of knotted AlphaFold-predicted models

The availability of 3D protein models is rapidly increasing with the development of structure prediction algorithms. With the expanding availability of data, new ways of analysis, especially topological analysis, of those predictions are becoming necessary. Here, we present the updated version of the AlphaKnot service that provides a straightforward way of analyzing structure topology. It was designed specifically to determine knot types of the predicted structure models, however, it can be used for all structures, including the ones solved experimentally. AlphaKnot 2.0 provides the user's ability to obtain the knowledge necessary to assess the topological correctness of the model. Both probabilistic and deterministic knot detection methods are available, together with various visualizations (including a trajectory of simplification steps to highlight the topological complexities). Moreover, the web server provides a list of proteins similar to the queried model within AlphaKnot's database and returns their knot types for direct comparison. We pre-calculated the topology of high-quality models from the AlphaFold Database (4th version) and there are now more than 680.000 knotted models available in the AlphaKnot database. AlphaKnot 2.0 is available at https://alphaknot.cent.uw.edu.pl/.

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.

Mathematical topology and geometry-based classification of tauopathies

Neurodegenerative diseases, like Alzheimer's, are associated with the presence of neurofibrillary lesions formed by tau protein filaments in the cerebral cortex. While it is known that different morphologies of tau filaments characterize different neurodegenerative diseases, there are few metrics of global and local structure complexity that enable to quantify their structural diversity rigorously. In this manuscript, we employ for the first time mathematical topology and geometry to classify neurodegenerative diseases by using cryo-electron microscopy structures of tau filaments that are available in the Protein Data Bank. By employing mathematical topology metrics (Gauss linking integral, writhe and second Vassiliev measure) we achieve a consistent, but more refined classification of tauopathies, than what was previously observed through visual inspection. Our results reveal a hierarchy of classification from global to local topology and geometry characteristics. In particular, we find that tauopathies can be classified with respect to the handedness of their global conformations and the handedness of the relative orientations of their repeats. Progressive supranuclear palsy is identified as an outlier, with a more complex structure than the rest, reflected by a small, but observable knotoid structure (a diagrammatic structure representing non-trivial topology). This topological characteristic can be attributed to a pattern in the beginning of the R3 repeat that is present in all tauopathies but at different extent. Moreover, by comparing single filament to paired filament structures within tauopathies we find a consistent change in the side-chain orientations with respect to the alpha carbon atoms at the area of interaction.

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.

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

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

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

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

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.

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.

A Marcus-Type Inverted Region in the Translocation Kinetics of a Knotted Protein

Knotted proteins are rare but important species, yet how their complex topologies affect their physical properties is not fully understood. Here we combine single molecule nanopore experiments and all-atom MD simulations to study the electric-field-driven unfolding during the translocation through a model pore of individual protein knots important for methylating tRNA. One of these knots shows an unusual behavior that resembles the behavior of electrons hopping between two potential surfaces: as the electric potential driving the translocation reaction is increased, the rate eventually plateaus or slows back down in the "Marcus inverted regime". Our results shed light on the influence of topology in knotted proteins on their forced translocation through a pore connecting two electrostatic potential wells.

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.

De novo design of knotted tandem repeat proteins

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

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.

Nucleolar Essential Protein 1 (Nep1): Elucidation of enzymatic catalysis mechanism by molecular dynamics simulation and quantum mechanics study

The Nep1 protein is essential for the formation of eukaryotic and archaeal small ribosomal subunits, and it catalyzes the site-directed SAM-dependent methylation of pseudouridine (Ψ) during pre-rRNA processing. It possesses a non-trivial topology, namely, a 31 knot in the active site. Here, we address the issue of seemingly unfeasible deprotonation of Ψ in Nep1 active site by a distant aspartate residue (D101 in S. cerevisiae), using a combination of bioinformatics, computational, and experimental methods. We identified a conserved hydroxyl-containing amino acid (S233 in S. cerevisiae, T198 in A. fulgidus) that may act as a proton-transfer mediator. Molecular dynamics simulations, based on the crystal structure of S. cerevisiae, and on a complex generated by molecular docking in A. fulgidus, confirmed that this amino acid can shuttle protons, however, a water molecule in the active site may also serve this role. Quantum-chemical calculations based on density functional theory and the cluster approach showed that the water-mediated pathway is the most favorable for catalysis. Experimental kinetic and mutational studies reinforce the requirement for the aspartate D101, but not S233. These findings provide insight into the catalytic mechanisms underlying proton transfer over extended distances and comprehensively elucidate the mode of action of Nep1.

Elucidation of the folding pathway of a circular permutant of topologically knotted YbeA by tryptophan substitutions

CP74 is an engineered circular permutant of a deep trefoil knotted SpoU-TrmD (SPOUT) RNA methyl transferase protein YbeA from E. coli. We have previously established that the circular permutation unties the knotted topology of YbeA and CP74 forms a domain-swapped dimer with a large dimeric interface of ca. 4600 Å². To understand the impact of domain-swapping and the newly formed hinge region joining the two folded domains on the folding and stability of CP74, the five equally spaced tryptophan residues were individually substituted into phenylalanine to monitor their conformational and stability changes by a battery of biophysical tools. Far-UV circular dichroism, intrinsic fluorescence, and small-angle X-ray scattering dictated minimal global conformational perturbations to the native structures in the tryptophan variants. The structures of the tryptophan variants also showed the conservation of the domain-swapped ternary structure with the exception that the W72F exhibited significant asymmetry in the α-helix 5. Comparative global thermal and chemical stability analyses indicated the pivotal role of W100 in the folding of CP74 followed by W19 and W72. Solution-state NMR spectroscopy and hydrogen-deuterium exchange mass spectrometry further revealed the accumulation of a native-like intermediate state in which the hinge region made important contributions to maintain the domain-swapped ternary structure of CP74.

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.

Homology of homologous knotted proteins

Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here, we develop a mathematical pipeline that systematically analyses protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in contexts where standard knot theoretic tools fail.

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.

On the Classification of Polyhedral Links

Knots and links are ubiquitous in chemical systems. Their structure can be responsible for a variety of physical and chemical properties, making them very important in materials development. In this article, we analyze the topological structures of interlocking molecules composed of metal-peptide rings using the concept of polyhedral links. To that end, we discuss the topological classification of alternating polyhedral links.

Using steered molecular dynamic tension for assessing quality of computational protein structure models

The native structures of proteins, except for notable exceptions of intrinsically disordered proteins, in general take their most stable conformation in the physiological condition to maintain their structural framework so that their biological function can be properly carried out. Experimentally, the stability of a protein can be measured by several means, among which the pulling experiment using the atomic force microscope (AFM) stands as a unique method. AFM directly measures the resistance from unfolding, which can be quantified from the observed force-extension profile. It has been shown that key features observed in an AFM pulling experiment can be well reproduced by computational molecular dynamics simulations. Here, we applied computational pulling for estimating the accuracy of computational protein structure models under the hypothesis that the structural stability would positively correlated with the accuracy, i.e. the closeness to the native, of a model. We used in total 4929 structure models for 24 target proteins from the Critical Assessment of Techniques of Structure Prediction (CASP) and investigated if the magnitude of the break force, that is, the force required to rearrange the model's structure, from the force profile was sufficient information for selecting near-native models. We found that near-native models can be successfully selected by examining their break forces suggesting that high break force indeed indicates high stability of models. On the other hand, there were also near-native models that had relatively low peak forces. The mechanisms of the stability exhibited by the break forces were explored and discussed.

AlphaKnot: server to analyze entanglement in structures predicted by AlphaFold methods

AlphaKnot is a server that measures entanglement in AlphaFold-solved protein models while considering pLDDT confidence values. AlphaKnot has two main functions: (i) providing researchers with a webserver for analyzing knotting in their own AlphaFold predictions and (ii) providing a database of knotting in AlphaFold predictions from the 21 proteomes for which models have been published prior to 2022. The knotting is defined in a probabilistic fashion. The knotting complexity of proteins is presented in the form of a matrix diagram which shows users the knot type for the entire polypeptide chain and for each of its subchains. The dominant knot types as well as the computed locations of the knot cores (i.e. minimal portions of protein backbones that form a given knot type) are shown for each protein structure. Based mainly on the pLDDT confidence values, entanglements are classified as Knots, Unsure, and Artifacts. The database portion of the server can be used, for example, to examine protein geometry and entanglement-function correlations, as a reference set for protein modeling, and for facilitating evolutional studies. The AlphaKnot server can be found at https://alphaknot.cent.uw.edu.pl/.

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.

Slipknot or Crystallographic Error: A Computational Analysis of the Plasmodium falciparum DHFR Structural Folds

The presence of protein structures with atypical folds in the Protein Data Bank (PDB) is rare and may result from naturally occurring knots or crystallographic errors. Proper characterisation of such folds is imperative to understanding the basis of naturally existing knots and correcting crystallographic errors. If left uncorrected, such errors can frustrate downstream experiments that depend on the structures containing them. An atypical fold has been identified in P. falciparum dihydrofolate reductase (PfDHFR) between residues 20–51 (loop 1) and residues 191–205 (loop 2). This enzyme is key to drug discovery efforts in the parasite, necessitating a thorough characterisation of these folds. Using multiple sequence alignments (MSA), a unique insert was identified in loop 1 that exacerbates the appearance of the atypical fold-giving it a slipknot-like topology. However, PfDHFR has not been deposited in the knotted proteins database, and processing its structure failed to identify any knots within its folds. The application of protein homology modelling and molecular dynamics simulations on the DHFR domain of P. falciparum and those of two other organisms (E. coli and M. tuberculosis) that were used as molecular replacement templates in solving the PfDHFR structure revealed plausible unentangled or open conformations of these loops. These results will serve as guides for crystallographic experiments to provide further insights into the atypical folds identified.

PDBe-KB consortium, Varadi M, Anyango S, Armstrong D, Berrisford J, Choudhary P, Deshpande M, Nadzirin N, Nair SS, Pravda L, Tanweer A, Al-Lazikani B, Andreini C, Barton GJ, Bednar D, Berka K, Blundell T, Brock KP, Carazo JM, Damborsky J, David A, Dey S, Dunbrack R, Recio JF, Fraternali F, Gibson T, Helmer-Citterich M, Hoksza D, Hopf T, Jakubec D, Kannan N, Krivak R, Kumar M, Levy ED, London N, Macias JR, Srivatsan MM, Marks DS, Martens L, McGowan SA, McGreig JE, Modi V, Parra RG, Pepe G, Piovesan D, Prilusky J, Putignano V, Radusky LG, Ramasamy P, Rausch AO, Reuter N, Rodriguez LA, Rollins NJ, Rosato A, Rubach P, Serrano L, Singh G, Skoda P, Sorzano COS, Stourac J, Sulkowska JI, Svobodova R, Tichshenko N, Tosatto SCE, Vranken W, Wass MN, Xue D, Zaidman D, Thornton J, Sternberg M, Orengo C, Velankar S. PDBe-KB: collaboratively defining the biological context of structural data

The Protein Data Bank in Europe - Knowledge Base (PDBe-KB, https://pdbe-kb.org) is an open collaboration between world-leading specialist data resources contributing functional and biophysical annotations derived from or relevant to the Protein Data Bank (PDB). The goal of PDBe-KB is to place macromolecular structure data in their biological context by developing standardised data exchange formats and integrating functional annotations from the contributing partner resources into a knowledge graph that can provide valuable biological insights. Since we described PDBe-KB in 2019, there have been significant improvements in the variety of available annotation data sets and user functionality. Here, we provide an overview of the consortium, highlighting the addition of annotations such as predicted covalent binders, phosphorylation sites, effects of mutations on the protein structure and energetic local frustration. In addition, we describe a library of reusable web-based visualisation components and introduce new features such as a bulk download data service and a novel superposition service that generates clusters of superposed protein chains weekly for the whole PDB archive.

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.

Channels with Helical Modulation Display Stereospecific Sensitivity for Chiral Superstructures

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

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.

Metal-Peptide Nonafoil Knots and Decafoil Supercoils

Despite the frequent occurrence of knotted frameworks in protein structures, the latent potential of peptide strands to form entangled structures is rarely discussed in peptide chemistry. Here we report the construction of highly entangled molecular topologies from Ag(I) ions and tripeptide ligands. The efficient entanglement of metal-peptide strands and the wide scope for design of the amino acid side chains in these ligands enabled the construction of metal-peptide 9₁ torus knots and 10₁² torus links. Moreover, steric control of the peptide side chain induced ring opening and twisting of the torus framework, which resulted in an infinite toroidal supercoil nanostructure.

Single-Molecule Force Spectroscopy of Protein Folding

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

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.

"Protein" no longer means what it used to

Every biologist knows that the word protein describes a group of macromolecules essential to sustain life on Earth. As biologists, we are invariably trained under a protein paradigm established since the early twentieth century. However, in recent years, the term protein unveiled itself as an euphemism to describe the overwhelming heterogeneity of these compounds. Most of our current studies are targeted on carefully selected subsets of proteins, but we tend to think and write about these as representative of the whole population. Here we discuss how seeking for universal definitions and general rules in any arbitrarily segmented study would be misleading about the conclusions. Of course, it is not our purpose to discourage the use of the word protein. Instead, we suggest to embrace the extended universe of proteins to reach a deeper understanding of their full potential, realizing that the term encompasses a group of molecules very heterogeneous in terms of size, shape, chemistry and functions, i.e. the term protein no longer means what it used to.

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.

Learning to unknot

We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an algorithm to randomly generate N-crossing braids and their knot closures and discussing the induced prior on the distribution of knots, we apply binary classification to the UNKNOT decision problem. We find that the Reformer and shared-QK Transformer network architectures outperform fully-connected networks, though all perform at ≳ 95% accuracy. Perhaps surprisingly, we find that accuracy increases with the length of the braid word, and that the networks learn a direct correlation between the confidence of their predictions and the degree of the Jones polynomial. Finally, we utilize reinforcement learning (RL) to find sequences of Markov moves and braid relations that simplify knots and can identify unknots by explicitly giving the sequence of unknotting actions. Trust region policy optimization (TRPO) performs consistently well, reducing ≳ 80% of the unknots with up to 96 crossings we tested to the empty braid word, and thoroughly outperformed other RL algorithms and random walkers. Studying these actions, we find that braid relations are more useful in simplifying to the unknot than one of the Markov moves.

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.