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.

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.

Molecule Edit Graph Attention Network: Modeling Chemical Reactions as Sequences of Graph Edits.

The central challenge in automated synthesis planning is to be able to generate and predict outcomes of a diverse set of chemical reactions. In particular, in many cases, the most likely synthesis pathway cannot be applied due to additional constraints, which requires proposing alternative chemical reactions. With this in mind, we present Molecule Edit Graph Attention Network (MEGAN), an end-to-end encoder-decoder neural model. MEGAN is inspired by models that express a chemical reaction as a sequence of graph edits, akin to the arrow pushing formalism. We extend this model to retrosynthesis prediction (predicting substrates given the product of a chemical reaction) and scale it up to large data sets. We argue that representing the reaction as a sequence of edits enables MEGAN to efficiently explore the space of plausible chemical reactions, maintaining the flexibility of modeling the reaction in an end-to-end fashion and achieving state-of-the-art accuracy in standard benchmarks. Code and trained models are made available online at https://github.com/molecule-one/megan.

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

The KnotProt 2.0 database (the updated version of the KnotProt database) collects information about proteins which form knots and other entangled structures. New features in KnotProt 2.0 include the characterization of both probabilistic and deterministic entanglements which can be formed by disulfide bonds and interactions via ions, a refined characterization of entanglement in terms of knotoids, the identification of the so-called cysteine knots, the possibility to analyze all or a non-redundant set of proteins, and various technical updates. The KnotProt 2.0 database classifies all entangled proteins, represents their complexity in the form of a knotting fingerprint, and presents many biological and geometrical statistics based on these results. Currently the database contains >2000 entangled structures, and it regularly self-updates based on proteins deposited in the Protein Data Bank (PDB).

PyLink: a PyMOL plugin to identify links.

SUMMARY: Links are generalization of knots, that consist of several components. They appear in proteins, peptides and other biopolymers with disulfide bonds or ions interactions giving rise to the exceptional stability. Moreover because of this stability such biopolymers are the target of commercial and medical use (including anti-bacterial and insecticidal activity). Therefore, topological characterization of such biopolymers, not only provides explanation of their thermodynamical or mechanical properties, but paves the way to design templates in pharmaceutical applications. However, distinction between links and trivial topology is not an easy task. Here, we present PyLink-a PyMOL plugin suited to identify three types of links and perform comprehensive topological analysis of proteins rich in disulfide or ion bonds. PyLink can scan for the links automatically, or the user may specify their own components, including closed loops with several bridges and ion interactions. This creates the possibility of designing new biopolymers with desired properties. AVAILABILITY AND IMPLEMENTATION: The PyLink plugin, manual and tutorial videos are available at http://pylink.cent.uw.edu.pl.

Topological knots and links in proteins.

Twenty years after their discovery, knots in proteins are now quite well understood. They are believed to be functionally advantageous and provide extra stability to protein chains. In this work, we go one step further and search for links-entangled structures, more complex than knots, which consist of several components. We derive conditions that proteins need to meet to be able to form links. We search through the entire Protein Data Bank and identify several sequentially nonhomologous chains that form a Hopf link and a Solomon link. We relate topological properties of these proteins to their function and stability and show that the link topology is characteristic of eukaryotes only. We also explain how the presence of links affects the folding pathways of proteins. Finally, we define necessary conditions to form Borromean rings in proteins and show that no structure in the Protein Data Bank forms a link of this type.

GapRepairer: a server to model a structural gap and validate it using topological analysis.

Motivation: Over 25% of protein structures possess unresolved fragments. On the other hand, approximately 6% of protein chains have non-trivial topology (and form knots, slipknots, lassos and links). As the topology is fundamental for the proper function of proteins, modeling of topologically correct structures is decisive in various fields, including biophysics, biotechnology and molecular biology. However, none of the currently existing tools take into account the topology of the model and those which could be modified to include topology, demand experience in bioinformatics, protein topology and knot theory. Results: In this work, we present the GapRepairer-the server that fills the gap in the spectrum of structure modeling methods. Its easy and intuitive interface offers the power of Modeller homology modeling to many non-experts in the field. This server determines the topology of templates and predicted structures. Such information when possible is used by the server to suggest the best model, or it can be used by the user to score models or to design artificially (dis)entangled structures. Availability and implementation: GapRepairer server along with tutorials, usage notes, movies and the database of already repaired structures is available at http://gaprepairer.cent.uw.edu.pl. Supplementary information: Supplementary data are available at Bioinformatics online.

The exclusive effects of chaperonin on the behavior of proteins with 52 knot.

The folding of proteins with a complex knot is still an unresolved question. Based on representative members of Ubiquitin C-terminal Hydrolases (UCHs) that contain the 52 knot in the native state, we explain how UCHs are able to unfold and refold in vitro reversibly within the structure-based model. In particular, we identify two, topologically different folding/unfolding pathways and corroborate our results with experiment, recreating the chevron plot. We show that confinement effect of chaperonin or weak crowding greatly facilitates folding, simultaneously slowing down the unfolding process of UCHs, compared with bulk conditions. Finally, we analyze the existence of knots in the denaturated state of UCHs. The results of the work show that the crowded environment of the cell should have a positive effect on the kinetics of complex knotted proteins, especially when proteins with deeper knots are found in this family.

LinkProt: a database collecting information about biological links.

Protein chains are known to fold into topologically complex shapes, such as knots, slipknots or complex lassos. This complex topology of the chain can be considered as an additional feature of a protein, separate from secondary and tertiary structures. Moreover, the complex topology can be defined also as one additional structural level. The LinkProt database (http://linkprot.cent.uw.edu.pl) collects and displays information about protein links - topologically non-trivial structures made by up to four chains and complexes of chains (e.g. in capsids). The database presents deterministic links (with loops closed, e.g. by two disulfide bonds), links formed probabilistically and macromolecular links. The structures are classified according to their topology and presented using the minimal surface area method. The database is also equipped with basic tools which allow users to analyze the topology of arbitrary (bio)polymers.

PyLasso: a PyMOL plugin to identify lassos.

SUMMARY: Entanglement in macromolecules is an important phenomenon and a subject of multidisciplinary research. As recently discovered, around 4% of proteins form new entangled motifs, called lassos. Here we present the PyLasso-a PyMOL plugin to identify and analyse properties of lassos in proteins and other (bio)polymers, as well as in other biological, physical and mathematical systems. The PyLasso is a useful tool for all researchers working on modeling of macromolecules, structure prediction, properties of polymers, entanglement in fluids and fields, etc. AVAILABILITY AND IMPLEMENTATION: The PyLasso and tutorial videos are available at http://pylasso.cent.uw.edu.pl. CONTACT: jsulkowska@cent.uw.edu.pl.

LassoProt: server to analyze biopolymers with lassos.

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

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.

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

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

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

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

Genus trace reveals the topological complexity and domain structure of biomolecules.

The structure of bonds in biomolecules, such as base pairs in RNA chains or native interactions in proteins, can be presented in the form of a chord diagram. A given biomolecule is then characterized by the genus of an auxiliary two-dimensional surface associated to such a diagram. In this work we introduce the notion of the genus trace, which describes dependence of genus on the choice of a subchain of a given backbone chain. We find that the genus trace encodes interesting physical and biological information about a given biomolecule and its three dimensional structural complexity; in particular it gives a way to quantify how much more complicated a biomolecule is than its nested secondary structure alone would indicate. We illustrate this statement in many examples, involving both RNA and protein chains. First, we conduct a survey of all published RNA structures with better than 3 Å resolution in the PDB database, and find that the genus of natural structural RNAs has roughly linear dependence on their length. Then, we show that the genus trace captures properties of various types of base pairs in RNA, and enables the identification of the domain structure of a ribosome. Furthermore, we find that not only does the genus trace detect a domain structure, but it also predicts a cooperative folding pattern in multi-domain proteins. The genus trace turns out to be a useful and versatile tool, with many potential applications.

To Tie or Not to Tie? That Is the Question.

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

In Search of Functional Advantages of Knots in Proteins.

We analysed the structure of deeply knotted proteins representing three unrelated families of knotted proteins. We looked at the correlation between positions of knotted cores in these proteins and such local structural characteristics as the number of intra-chain contacts, structural stability and solvent accessibility. We observed that the knotted cores and especially their borders showed strong enrichment in the number of contacts. These regions showed also increased thermal stability, whereas their solvent accessibility was decreased. Interestingly, the active sites within these knotted proteins preferentially located in the regions with increased number of contacts that also have increased thermal stability and decreased solvent accessibility. Our results suggest that knotting of polypeptide chains provides a favourable environment for the active sites observed in knotted proteins. Some knotted proteins have homologues without a knot. Interestingly, these unknotted homologues form local entanglements that retain structural characteristics of the knotted cores.

Complex lasso: new entangled motifs in proteins.

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