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.

AlphaFold predicts novel human proteins with knots.

The fact that proteins can have their chain formed in a knot is known for almost 30 years. However, as they are not common, only a fraction of such proteins is available in the Protein Data Bank. It was not possible to assess their importance and versatility up until now because we did not have access to the whole proteome of an organism, let alone a human one. The arrival of efficient machine learning methods for protein structure prediction, such as AlphaFold and RoseTTaFold, changed that. We analyzed all proteins from the human proteome (over 20,000) determined with AlphaFold in search for knots and found them in less than 2% of the structures. Using a variety of methods, including homolog search, clustering, quality assessment, and visual inspection, we determined the nature of each of the knotted structures and classified it as either knotted, potentially knotted, or an artifact, and deposited all of them in a database available at: https://knotprot.cent.uw.edu.pl/alphafold. Overall, we found 51 credible knotted proteins (0.2% of human proteome). The set of potentially knotted structures includes a new complex type of a knot not reported in proteins yet. That knot type, denoted 63 in mathematical notation, would necessitate a more complex folding path than any knotted protein characterized to date.

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/.

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.

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.

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.

Defining and detecting links in chromosomes.

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

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.

KnotGenome: a server to analyze entanglements of chromosomes.

The KnotGenome server enables the topological analysis of chromosome model data using three-dimensional coordinate files of chromosomes as input. In particular, it detects prime and composite knots in single chromosomes, and links between chromosomes. The knotting complexity of the chromosome is presented in the form of a matrix diagram that reveals the knot type of the entire polynucleotide chain and of each of its subchains. Links are determined by means of the Gaussian linking integral and the HOMFLY-PT polynomial. Entangled chromosomes are presented graphically in an intuitive way. It is also possible to relax structure with short molecular dynamics runs before the analysis. KnotGenome is freely available at http://knotgenom.cent.uw.edu.pl/.

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.

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.

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.

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.

KnotProt: a database of proteins with knots and slipknots.

The protein topology database KnotProt, http://knotprot.cent.uw.edu.pl/, collects information about protein structures with open polypeptide chains forming knots or slipknots. The knotting complexity of the cataloged proteins is presented in the form of a matrix diagram that shows users the knot type of the entire polypeptide chain and of each of its subchains. The pattern visible in the matrix gives the knotting fingerprint of a given protein and permits users to determine, for example, the minimal length of the knotted regions (knot's core size) or the depth of a knot, i.e. how many amino acids can be removed from either end of the cataloged protein structure before converting it from a knot to a different type of knot. In addition, the database presents extensive information about the biological functions, families and fold types of proteins with non-trivial knotting. As an additional feature, the KnotProt database enables users to submit protein or polymer chains and generate their knotting fingerprints.