Wanda Niemyska

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 (knots 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.

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.

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.

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.

Laws of Contraposition and Law of Importation for Probabilistic Implications and Probabilistic S-implications

Recently, Grzegorzewski [5-7] introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In the same article [7] author gives a motivation to his idea and indicates some interesting connections between new families of implications and the dependence structure of the underlying environment. In this paper the laws of contraposition and the law of importation are studied for these families of fuzzy implications.

Properties of the probabilistic implications and S-implications

Recently, Grzegorzewski (2011) introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In his previous articles author gave a motivation to his idea and indicates some interesting connections between new families of multivalued implications and the dependence structure of the underlying environment. In this paper the laws of contraposition, the law of importation and T-conditionality are studied for these families of fuzzy implications. Furthermore, we discuss the intersections of both new families of implications with R-implications, (S,N)-implications and QL-operations.

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.

Fuzzy implications based on semicopulas

Abstract Probabilistic implications and probabilistic S-implications, introduced recently by Grzegorzewski, have attracted attention of researchers engaged in fuzzy implications. Since these two families are based on copulas, they form a kind of bridge between probability theory and fuzzy logic. In this paper we generalize the aforementioned two classes and propose a new approach for constructing fuzzy implications which combines a given a priori fuzzy implication and a semicopula.

On a functional equation related to distributivity of fuzzy implications

Recently in some considerations connected with the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared f(min(x + y, a)) = min(f(x) + f(y)m b). In the current paper we consider a generalized version of this equation, namely the equation f(m₁(x + y)) = m₂(f(x) + f(y)), where m₁, m₂ are functions defined on some intervals of ℝ satisfying additional assumptions. We analyze the cases when m₂ is injective and when m₂ is not injective.

Sheffer Stroke Fuzzy Implications.

A new family of fuzzy implications, motivated by classic Sheffer stroke operator, is introduced. Sheffer stroke, which is a negation of a conjunction and is called NAND as well, is one of the two operators that can be used by itself, without any other logical operators, to constitute a logical formal system. Classical implication can be presented just by Sheffer stroke operator in two ways which leads to two new families of fuzzy implication functions. It turns out that one of them is mainly a subclass of QL-operations, while the other one, called in our paper as SS\(_{qq}\)-implications, is independent of other well-known families of fuzzy implications. Basic properties of Sheffer stroke implications are also analysed.