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/.

RiFNet: Automated rib fracture detection in postmortem computed tomography.

Imaging techniques are widely used for medical diagnostics. In some cases, a lack of medical practitioners who can manually analyze the images can lead to a bottleneck. Consequently, we developed a custom-made convolutional neural network (RiFNet = Rib Fracture Network) that can detect rib fractures in postmortem computed tomography. In a retrospective cohort study, we retrieved PMCT data from 195 postmortem cases with rib fractures from July 2017 to April 2018 from our database. The computed tomography data were prepared using a plugin in the commercial imaging software Syngo.via whereby the rib cage was unfolded on a single-in-plane image reformation. Out of the 195 cases, a total of 585 images were extracted and divided into two groups labeled "with" and "without" fractures. These two groups were subsequently divided into training, validation, and test datasets to assess the performance of RiFNet. In addition, we explored the possibility of applying transfer learning techniques on our dataset by choosing two independent noncommercial off-the-shelf convolutional neural network architectures (ResNet50 V2 and Inception V3) and compared the performances of those two with RiFNet. When using pre-trained convolutional neural networks, we achieved an F1 score of 0.64 with Inception V3 and an F1 score of 0.61 with ResNet50 V2. We obtained an average F1 score of 0.91 ± 0.04 with RiFNet. RiFNet is efficient in detecting rib fractures on postmortem computed tomography. Transfer learning techniques are not necessarily well adapted to make classifications in postmortem computed tomography.

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).

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.

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.

Knotting fingerprints resolve knot complexity and knotting pathways in ideal knots.

We use disk matrices to define knotting fingerprints that provide fine-grained insights into the local knotting structure of ideal knots. These knots have been found to have spatial properties that highly correlate with those of interesting macromolecules. From this fine structure and an analysis of the associated planar graph, one can define a measure of knot complexity using the number of independent unknotting pathways from the global knot type as the knot is trimmed progressively to a short arc unknot. A specialization of the Cheeger constant provides a measure of constraint on these independent unknotting pathways. Furthermore, the structure of the knotting fingerprint supports a comparison of the tight knot pathways to the unconstrained unknotting pathways of comparable length.

Subknots in ideal knots, random knots, and knotted proteins.

We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protocol. We characterize the sets of observed subknot types in global knots taking energy-minimized shapes such as KnotPlot configurations and ideal geometric configurations. We compare the sets of observed subknots to knot types obtained by changing crossings in the classical prime knot diagrams. Building upon this analysis, we study the sets of subknots in random configurations of corresponding knot types. In many of the knot types we analyzed, the sets of subknots from the ideal geometric configurations are found in each of the hundreds of random configurations of the same global knot type. We also compare the sets of subknots observed in open protein knots with the subknots observed in the ideal configurations of the corresponding knot type. This comparison enables us to explain the specific dispositions of subknots in the analyzed protein knots.

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.

Knotting pathways in proteins.

Most proteins, in order to perform their biological function, have to fold to a compact native state. The increasing number of knotted and slipknotted proteins identified suggests that proteins are able to manoeuvre around topological barriers during folding. In the present article, we review the current progress in elucidating the knotting process in proteins. Although we concentrate on theoretical approaches, where a knotted topology can be unambiguously detected, comparison with experiments is also reviewed. Numerical simulations suggest that the folding process for small knotted proteins is composed of twisted loop formation and then threading by either slipknot geometries or flipping. As the size of the knotted proteins increases, particularly for more deeply threaded termini, the prevalence of traps in the free energy landscape also increases. Thus, in the case of longer knotted and slipknotted proteins, the folding mechanism is probably supported by chaperones. Overall, results imply that knotted proteins can be folded efficiently and survive evolutionary pressure in order to perform their biological functions.

Identifying knots in proteins.

Polypeptide chains form open knots in many proteins. How these knotted proteins fold and finding the evolutionary advantage provided by these knots are among some of the key questions currently being studied in the protein folding field. The detection and identification of protein knots are substantial challenges. Different methods and many variations of them have been employed, but they can give different results for the same protein. In the present article, we review the various knot identification algorithms and compare their relative strengths when applied to the study of knots in proteins. We show that the statistical approach based on the uniform closure method is advantageous in comparison with other methods used to characterize protein knots.

Knot localization in proteins.

The backbones of proteins form linear chains. In the case of some proteins, these chains can be characterized as forming linear open knots. The knot type of the full chain reveals only limited information about the entanglement of the chain since, for example, subchains of an unknotted protein can form knots and subchains of a knotted protein can form different types of knots than the entire protein. To understand fully the entanglement within the backbone of a given protein, a complete analysis of the knotting within all of the subchains of that protein is necessary. In the present article, we review efforts to characterize the full knotting complexity within individual proteins and present a matrix that conveys information about various aspects of protein knotting. For a given protein, this matrix identifies the precise localization of knotted regions and shows the knot types formed by all subchains. The pattern in the matrix can be considered as a knotting fingerprint of that protein. We observe that knotting fingerprints of distantly related knotted proteins are strongly conserved during evolution and discuss how some characteristic motifs in the knotting fingerprints are related to the structure of the knotted regions and their possible biological role.

Sedimentation of macroscopic rigid knots and its relation to gel electrophoretic mobility of DNA knots.

We address the general question of the extent to which the hydrodynamic behaviour of microscopic freely fluctuating objects can be reproduced by macrosopic rigid objects. In particular, we compare the sedimentation speeds of knotted DNA molecules undergoing gel electrophoresis to the sedimentation speeds of rigid stereolithographic models of ideal knots in both water and silicon oil. We find that the sedimentation speeds grow roughly linearly with the average crossing number of the ideal knot configurations, and that the correlation is stronger within classes of knots. This is consistent with previous observations with DNA knots in gel electrophoresis.

All monofilament knots assume sliding conformation in vivo.

BACKGROUND: Surgeons are not always cognizant of the knots they tie. It has been thought that suture material does not determine what types of knots are tied. DESIGN: Experienced surgeons were asked to tie square monofilament knots that were salvaged and microscopically evaluated. RESULTS: Tightly cinched monofilament knots assume sliding conformation in vivo regardless of configuration, instrumentation, suture polymer, or surgeon. CONCLUSIONS: When monofilament suture is securely cinched in vivo, kinetic energy forces even a flat throw into sliding conformation. A well-crafted, intentionally sliding square knot appears compacter than any other monofilament knot.

Conservation of complex knotting and slipknotting patterns in proteins.

While analyzing all available protein structures for the presence of knots and slipknots, we detected a strict conservation of complex knotting patterns within and between several protein families despite their large sequence divergence. Because protein folding pathways leading to knotted native protein structures are slower and less efficient than those leading to unknotted proteins with similar size and sequence, the strict conservation of the knotting patterns indicates an important physiological role of knots and slipknots in these proteins. Although little is known about the functional role of knots, recent studies have demonstrated a protein-stabilizing ability of knots and slipknots. Some of the conserved knotting patterns occur in proteins forming transmembrane channels where the slipknot loop seems to strap together the transmembrane helices forming the channel.

Symmetry-breaking in cumulative measures of shapes of polymer models.

Using numerical simulations we investigate shapes of random equilateral open and closed chains, one of the simplest models of freely fluctuating polymers in a solution. We are interested in the 3D density distribution of the modeled polymers where the polymers have been aligned with respect to their three principal axes of inertia. This type of approach was pioneered by Theodorou and Suter in 1985. While individual configurations of the modeled polymers are almost always nonsymmetric, the approach of Theodorou and Suter results in cumulative shapes that are highly symmetric. By taking advantage of asymmetries within the individual configurations, we modify the procedure of aligning independent configurations in a way that shows their asymmetry. This approach reveals, for example, that the 3D density distribution for linear polymers has a bean shape predicted theoretically by Kuhn. The symmetry-breaking approach reveals complementary information to the traditional, symmetrical, 3D density distributions originally introduced by Theodorou and Suter.

Effect of knotting on polymer shapes and their enveloping ellipsoids.

We simulate freely jointed chains to investigate how knotting affects the overall shapes of freely fluctuating circular polymeric chains. To characterize the shapes of knotted polygons, we construct enveloping ellipsoids that minimize volume while containing the entire polygon. The lengths of the three principal axes of the enveloping ellipsoids are used to define universal size and shape descriptors analogous to the squared radius of gyration and the inertial asphericity and prolateness. We observe that polymeric chains forming more complex knots are more spherical and also more prolate than chains forming less complex knots with the same number of edges. We compare the shape measures, determined by the enveloping ellipsoids, with those based on constructing inertial ellipsoids and explain the differences between these two measures of polymer shape.

Bending modes of DNA directly addressed by cryo-electron microscopy of DNA minicircles.

We use cryo-electron microscopy (cryo-EM) to study the 3D shapes of 94-bp-long DNA minicircles and address the question of whether cyclization of such short DNA molecules necessitates the formation of sharp, localized kinks in DNA or whether the necessary bending can be redistributed and accomplished within the limits of the elastic, standard model of DNA flexibility. By comparing the shapes of covalently closed, nicked and gapped DNA minicircles, we conclude that 94-bp-long covalently closed and nicked DNA minicircles do not show sharp kinks while gapped DNA molecules, containing very flexible single-stranded regions, do show sharp kinks. We corroborate the results of cryo-EM studies by using Bal31 nuclease to probe for the existence of kinks in 94-bp-long minicircles.

Length of the tightest trefoil knot.

The physical sense of tight knots provided by the SONO algorithm is discussed. A method allowing one to predict their length is presented. An upper bound for the minimum length of a smooth trefoil knot is determined.

Role of flexibility in entanglement.

Entanglement is essential to the function of many physical systems. Flexibility and length determine the extent to which the system can become entangled. Given a perfectly flexible unit-radius tube, several researchers have studied the minimum length needed to tie different types of knots. Can one obtain the same configurations with less flexible tubing? Does more flexibility always yield tighter knots? We demonstrate a phase change in flexibility beyond which more flexibility adds very little entanglement. This level of flexibility is surprisingly low and appears to have a global bound. Since tensile strength and flexibility act inversely, this level of flexibility provides the maximal tensile strength for materials that need to pack tightly. This is a basic design principle that should be observable in nature.