Complex lasso: new entangled motifs in proteins

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/.

Information Connections among Multiple Investors: Evolutionary Local Patterns Revealed by Motifs

The concept of motifs provides a fresh perspective for studying local patterns, which is useful for understanding the essence of a network structure. However, few previous studies have focused on the evolutionary characteristics of weighted motifs while further considering participants’ differences. We study how information connections differ among multiple investors. The evolutionary 10-year trend of weighted 3-motifs in China’s energy stock markets is explored for the networks of co-holding behaviors among shareholders, who are classified as companies, funds and individuals. Our works allow us to detect the preferential local patterns distributed among different agents as their fluctuate involvement in networks. We find that the diversity of shareholders contributes to the statistical significance of local patterns, while homophily always exist among individuals. Modules of information connections are stable among reserved investors, which is especially apparent among companies. Individuals prefer to keep their connections with companies and funds. Unsteady modules happen owing to strengthen links among funds during the time that they are main participants in stock markets. More details about multiple investors informationally connected in evolutionary local patterns can be detected by our work.

To Tie or Not to Tie? That Is the Question

In this review, we provide an overview of entangled proteins. Around 6% of protein structures deposited in the PBD are entangled, forming knots, slipknots, lassos and links. We present theoretical methods and tools that enabled discovering and classifying such structures. We discuss the advantages and disadvantages of the non-trivial topology in proteins, based on available data about folding, stability, biological properties and evolutionary conservation. We also formulate intriguing and challenging questions on the border of biophysics, bioinformatics, biology and mathematics, which arise from the discovery of an entanglement in proteins. Finally, we discuss possible applications of entangled proteins in medicine and nanotechnology, such as the chance to design super stable proteins, whose stability could be controlled by chemical potential.

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.

Topological knots and links in proteins

Twenty years after their discovery, knots in proteins are now quite well understood. They are believed to be functionally advantageous and provide extra stability to protein chains. In this work, we go one step further and search for links-entangled structures, more complex than knots, which consist of several components. We derive conditions that proteins need to meet to be able to form links. We search through the entire Protein Data Bank and identify several sequentially nonhomologous chains that form a Hopf link and a Solomon link. We relate topological properties of these proteins to their function and stability and show that the link topology is characteristic of eukaryotes only. We also explain how the presence of links affects the folding pathways of proteins. Finally, we define necessary conditions to form Borromean rings in proteins and show that no structure in the Protein Data Bank forms a link of this type.

Proteins analysed as virtual knots

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the termini of the chain somehow. We introduce a new method for resolving knotting in open curves using virtual knots, which are a wider class of topological objects that do not require a classical closure and so naturally capture the topological ambiguity inherent in open curves. We describe the results of analysing proteins in the Protein Data Bank by this new scheme, recovering and extending previous knotting results, and identifying topological interest in some new cases. The statistics of virtual knots in protein chains are compared with those of open random walks and Hamiltonian subchains on cubic lattices, identifying a regime of open curves in which the virtual knotting description is likely to be important.

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.