Configuration Census. Topological Chirality and the New Combinatorial Invariants

In this paper we describe new mathematical methods which can be used to distinguish between configurations of knotted and/er linked chains in space and to determine their topological chirality. These new methods are primarily algebraic and combinatorial and are easily understood in a calculational context. The methods are presented and discussed through the study of several fundamental examples. The polynomial invariants are compared and the development of other sensitive algebraic-combinatorial invariants which may have important applications in chemistry is discussed. A table of the polynomials associated to the basic examples is given

Knots in knots: A study of classical knot diagrams

Looking at the structure of minimal prime knot presentations, one can notice that there are often, perhaps always, segments that present either the trefoil or the figure-eight knot. This note explores the question as to whether this is always the case, reporting on conversations with Jablan Slavik that began at a [Formula: see text] conference in Trieste, Italy and which never reached a conclusion. Evidence supporting this conjectured presence is reported and potential consequences are described

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 fingerprint

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

Dimethyl fumarate in patients admitted to hospital with COVID-19 (RECOVERY): a randomised, controlled, open-label, platform trial

Dimethyl fumarate (DMF) inhibits inflammasome-mediated inflammation and has been proposed as a treatment for patients hospitalised with COVID-19. This randomised, controlled, open-label platform trial (Randomised Evaluation of COVID-19 Therapy [RECOVERY]), is assessing multiple treatments in patients hospitalised for COVID-19 (NCT04381936, ISRCTN50189673). In this assessment of DMF performed at 27 UK hospitals, adults were randomly allocated (1:1) to either usual standard of care alone or usual standard of care plus DMF. The primary outcome was clinical status on day 5 measured on a seven-point ordinal scale. Secondary outcomes were time to sustained improvement in clinical status, time to discharge, day 5 peripheral blood oxygenation, day 5 C-reactive protein, and improvement in day 10 clinical status. Between 2 March 2021 and 18 November 2021, 713 patients were enroled in the DMF evaluation, of whom 356 were randomly allocated to receive usual care plus DMF, and 357 to usual care alone. 95% of patients received corticosteroids as part of routine care. There was no evidence of a beneficial effect of DMF on clinical status at day 5 (common odds ratio of unfavourable outcome 1.12; 95% CI 0.86-1.47; p = 0.40). There was no significant effect of DMF on any secondary outcome

Knots in knots: A study of classical knot diagrams

Looking at the structure of minimal prime knot presentations, one can notice that there are often, perhaps always, segments that present either the trefoil or the figure-eight knot. This note explores the question as to whether this is always the case, reporting on conversations with Jablan Slavik that began at a [Formula: see text] conference in Trieste, Italy and which never reached a conclusion. Evidence supporting this conjectured presence is reported and potential consequences are described