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.

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.

Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology.

Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process such as the topoisomerase-mediated preferential unknotting of DNA molecules. We model here topoisomerase-mediated passages of double-stranded DNA segments through each other using the formalism of grid diagrams. We show that this grid diagram-based modeling approach captures the essence of the preferential unknotting mechanism, based on topoisomerase selectivity of hooked DNA juxtapositions as the sites of intersegmental passages. We show that the grid diagram-based approach provides an important, new, and computationally convenient framework for investigating entanglement in biopolymers.

Chromatin Is Frequently Unknotted at the Megabase Scale.

Knots in the human genome would greatly impact diverse cellular processes ranging from transcription to gene regulation. To date, it has not been possible to directly examine the genome in vivo for the presence of knots. Recently, methods for serial fluorescent in situ hybridization have made it possible to measure the three-dimensional position of dozens of consecutive genomic loci in vivo. However, the determination of whether genomic trajectories are knotted remains challenging because small errors in the localization of a single locus can transform an unknotted trajectory into a highly knotted trajectory and vice versa. Here, we use stochastic closure analysis to determine if a genomic trajectory is knotted in the setting of experimental noise. We analyze 4727 deposited genomic trajectories of a 2-Mb-long chromatin interval from human chromosome 21. For 243 of these trajectories, their knottedness could be reliably determined despite the possibility of localization errors. Strikingly, in each of these 243 cases, the trajectory was unknotted. We note a potential source of bias insofar as knotted contours may be more difficult to reliably resolve. Nevertheless, our data are consistent with a model in which, at the scales probed, the human genome is often free of knots.

Closing the DNA replication cycle: from simple circular molecules to supercoiled and knotted DNA catenanes.

Due to helical structure of DNA, massive amounts of positive supercoils are constantly introduced ahead of each replication fork. Positive supercoiling inhibits progression of replication forks but various mechanisms evolved that permit very efficient relaxation of that positive supercoiling. Some of these mechanisms lead to interesting topological situations where DNA supercoiling, catenation and knotting coexist and influence each other in DNA molecules being replicated. Here, we first review fundamental aspects of DNA supercoiling, catenation and knotting when these qualitatively different topological states do not coexist in the same circular DNA but also when they are present at the same time in replicating DNA molecules. We also review differences between eukaryotic and prokaryotic cellular strategies that permit relaxation of positive supercoiling arising ahead of the replication forks. We end our review by discussing very recent studies giving a long-sought answer to the question of how slow DNA topoisomerases capable of relaxing just a few positive supercoils per second can counteract the introduction of hundreds of positive supercoils per second ahead of advancing replication forks.

Are TADs supercoiled?

Topologically associating domains (TADs) are megabase-sized building blocks of interphase chromosomes in higher eukaryotes. TADs are chromosomal regions with increased frequency of internal interactions. On average a pair of loci separated by a given genomic distance contact each other 2-3 times more frequently when they are in the same TAD as compared to a pair of loci located in two neighbouring TADs. TADs are also functional blocks of chromosomes as enhancers and their cognate promoters are normally located in the same TAD, even if their genomic distance from each other can be as large as a megabase. The internal structure of TADs, causing their increased frequency of internal interactions, is not established yet. We survey here experimental studies investigating presence of supercoiling in interphase chromosomes. We also review numerical simulation studies testing whether transcription-induced supercoiling of chromatin fibres can explain how TADs are formed and how they can assure very efficient interactions between enhancers and their cognate promoters located in the same TAD.

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).

Periodic Motion of Sedimenting Flexible Knots.

We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

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.

Two convergent pathways of DNA knotting in replicating DNA molecules as revealed by θ-curve analysis.

During DNA replication in living cells some DNA knots are inadvertently produced by DNA topoisomerases facilitating progression of replication forks. The types of DNA knots formed are conditioned by the 3D organization of replicating DNA molecules. Therefore, by characterizing formed DNA knots it is possible to infer the 3D arrangement of replicating DNA molecules. This topological inference method is highly developed for knotted DNA circles. However, partially replicated DNA molecules have the form of θ-curves. In this article, we use mathematical formalism of θ-curves to characterize the full possibilities of how knotting can occur during replication of DNA molecules in vivo. To do this, we reanalyze earlier experimental studies of knotted, partially replicated DNA molecules and the previously proposed pathway of their formation. We propose a general model of knotting in replication intermediates, and demonstrate that there is an additional, equally important, parallel knotting pathway that also explains how DNA topoisomerases can produce experimentally observed knotted θ-curves. Interestingly, both pathways require intertwining of freshly replicated sister duplexes (precatenanes).

Knoto-ID: a tool to study the entanglement of open protein chains using the concept of knotoids.

Summary: The backbone of most proteins forms an open curve. To study their entanglement, a common strategy consists in searching for the presence of knots in their backbones using topological invariants. However, this approach requires to close the curve into a loop, which alters the geometry of curve. Knoto-ID allows evaluating the entanglement of open curves without the need to close them, using the recent concept of knotoids which is a generalization of the classical knot theory to open curves. Knoto-ID can analyse the global topology of the full chain as well as the local topology by exhaustively studying all subchains or only determining the knotted core. Knoto-ID permits to localize topologically non-trivial protein folds that are not detected by informatics tools detecting knotted protein folds. Availability and implementation: Knoto-ID is written in C++ and includes R (www.R-project.org) scripts to generate plots of projections maps, fingerprint matrices and disk matrices. Knoto-ID is distributed under the GNU General Public License (GPL), version 2 or any later version and is available at https://github.com/sib-swiss/Knoto-ID. A binary distribution for Mac OS X, Linux and Windows with detailed user guide and examples can be obtained from https://www.vital-it.ch/software/Knoto-ID.

Chromatin Loop Extrusion and Chromatin Unknotting.

It has been a puzzle how decondensed interphase chromosomes remain essentially unknotted. The natural expectation is that in the presence of type II DNA topoisomerases that permit passages of double-stranded DNA regions through each other, all chromosomes should reach the state of topological equilibrium. The topological equilibrium in highly crowded interphase chromosomes forming chromosome territories would result in formation of highly knotted chromatin fibres. However, Chromosome Conformation Capture (3C) methods revealed that the decay of contact probabilities with the genomic distance in interphase chromosomes is practically the same as in the crumpled globule state that is formed when long polymers condense without formation of any knots. To remove knots from highly crowded chromatin, one would need an active process that should not only provide the energy to move the system from the state of topological equilibrium but also guide topoisomerase-mediated passages in such a way that knots would be efficiently unknotted instead of making the knots even more complex. We perform coarse-grained molecular dynamics simulations of the process of chromatin loop extrusion involving knotted and catenated chromatin fibres to check whether chromatin loop extrusion may be involved in active unknotting of chromatin fibres. Our simulations show that the process of chromatin loop extrusion is ideally suited to actively unknot, decatenate and demix chromatin fibres in interphase chromosomes.

Transcription-induced supercoiling as the driving force of chromatin loop extrusion during formation of TADs in interphase chromosomes.

Using molecular dynamics simulations, we show here that growing plectonemes resulting from transcription-induced supercoiling have the ability to actively push cohesin rings along chromatin fibres. The pushing direction is such that within each topologically associating domain (TAD) cohesin rings forming handcuffs move from the source of supercoiling, constituted by RNA polymerase with associated DNA topoisomerase TOP1, towards borders of TADs, where supercoiling is released by topoisomerase TOPIIB. Cohesin handcuffs are pushed by continuous flux of supercoiling that is generated by transcription and is then progressively released by action of TOPIIB located at TADs borders. Our model explains what can be the driving force of chromatin loop extrusion and how it can be ensured that loops grow quickly and in a good direction. In addition, the supercoiling-driven loop extrusion mechanism is consistent with earlier explanations proposing why TADs flanked by convergent CTCF binding sites form more stable chromatin loops than TADs flanked by divergent CTCF binding sites. We discuss the role of supercoiling in stimulating enhancer promoter contacts and propose that transcription of eRNA sends the first wave of supercoiling that can activate mRNA transcription in a given TAD.

Transcription-induced supercoiling explains formation of self-interacting chromatin domains in S. pombe.

The question of how self-interacting chromatin domains in interphase chromosomes are structured and generated dominates current discussions on eukaryotic chromosomes. Numerical simulations using standard polymer models have been helpful in testing the validity of various models of chromosome organization. Experimental contact maps can be compared with simulated contact maps and thus verify how good is the model. With increasing resolution of experimental contact maps, it became apparent though that active processes need to be introduced into models to recapitulate the experimental data. Since transcribing RNA polymerases are very strong molecular motors that induce axial rotation of transcribed DNA, we present here models that include such rotational motors. We also include into our models swivels and sites for intersegmental passages that account for action of DNA topoisomerases releasing torsional stress. Using these elements in our models, we show that transcription-induced supercoiling generated in the regions with divergent-transcription and supercoiling relaxation occurring between these regions are sufficient to explain formation of self-interacting chromatin domains in chromosomes of fission yeast (S. pombe).

Molecular Dynamics Simulation of Supercoiled, Knotted, and Catenated DNA Molecules, Including Modeling of Action of DNA Gyrase.

A detailed protocol of molecular dynamics simulations of supercoiled DNA molecules that can be in addition knotted or catenated is described. We also describe how to model ongoing action of DNA gyrase that introduces negative supercoing into DNA molecules. The protocols provide detailed instructions about model parameters, equations of used potentials, simulation, and visualization. Implementation of the model into a frequently used molecular dynamics simulation environment, ESPResSo, is shown step by step.

Studies of global and local entanglements of individual protein chains using the concept of knotoids.

We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contrast to knot diagrams which represent projections of closed curves in 3D space. The intrinsic difference with classical knot theory is that the generalization provided by knotoids admits non-trivial topological entanglement of the open curves provided that their geometry is frozen as it is the case for crystallized proteins. Consequently, our approach doesn't require the closure of chains into loops which implies that the geometry of analysed chains does not need to be changed by closure in order to characterize their topology. Our study revealed that the knotoid approach detects protein regions that were classified earlier as knotted and also new, topologically interesting regions that we classify as pre-knotted.

Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids.

In this paper we introduce a method that offers a detailed overview of the entanglement of an open protein chain. Further, we present a purely topological model for classifying open protein chains by also taking into account any bridge involving the backbone. To this end, we implemented the concepts of planar knotoids and bonded knotoids. We show that the planar knotoids technique provides more refined information regarding the knottedness of a protein when compared to established methods in the literature. Moreover, we demonstrate that our topological model for bonded proteins is robust enough to distinguish all types of lassos in proteins.

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.

How topoisomerase IV can efficiently unknot and decatenate negatively supercoiled DNA molecules without causing their torsional relaxation.

Freshly replicated DNA molecules initially form multiply interlinked right-handed catenanes. In bacteria, these catenated molecules become supercoiled by DNA gyrase before they undergo a complete decatenation by topoisomerase IV (Topo IV). Topo IV is also involved in the unknotting of supercoiled DNA molecules. Using Metropolis Monte Carlo simulations, we investigate the shapes of supercoiled DNA molecules that are either knotted or catenated. We are especially interested in understanding how Topo IV can unknot right-handed knots and decatenate right-handed catenanes without acting on right-handed plectonemes in negatively supercoiled DNA molecules. To this end, we investigate how the topological consequences of intersegmental passages depend on the geometry of the DNA-DNA juxtapositions at which these passages occur. We observe that there are interesting differences between the geometries of DNA-DNA juxtapositions in the interwound portions and in the knotted or catenated portions of the studied molecules. In particular, in negatively supercoiled, multiply interlinked, right-handed catenanes, we detect specific regions where DNA segments belonging to two freshly replicated sister DNA molecules form left-handed crossings. We propose that, due to its geometrical preference to act on left-handed crossings, Topo IV can specifically unknot supercoiled DNA, as well as decatenate postreplicative catenanes, without causing their torsional relaxation.