A circuit topology approach to categorizing changes in biomolecular structure

The biological world is composed of folded linear molecules of bewildering topological complexity and diversity. The topology of folded biomolecules such as proteins and ribonucleic acids is often subject to change during biological processes. Despite intense research, we lack a solid mathematical framework that summarizes these operations in a principled manner. Circuit topology, which formalizes the arrangements of intramolecular contacts, serves as a general mathematical framework to analyze the topological characteristics of folded linear molecules. In this work, we translate familiar molecular operations in biology, such as duplication, permutation, and elimination of contacts, into the language of circuit topology. We show that for such operations there are corresponding matrix representations as well as basic rules that serve as a foundation for understanding these operations within the context of a coherent algebraic framework. We present several biological examples and provide a simple computational framework for creating and analyzing the circuit diagrams of proteins and nucleic acids. We expect our study and future developments in this direction to facilitate a deeper understanding of natural molecular processes and to provide guidance to engineers for generating complex polymeric materials

Decoding chirality in circuit topology of a self entangled chain through braiding

Circuit topology employs fundamental units of entanglement, known as soft contacts, for constructing knots from the bottom up, utilising circuit topology relations, namely parallel, series, cross, and concerted relations. In this article, we further develop this approach to facilitate the analysis of chirality, which is a significant quantity in polymer chemistry. To achieve this, we translate the circuit topology approach to knot engineering into a braid-theoretic framework. This enables us to calculate the Jones polynomial for all possible binary combinations of contacts in cross or concerted relations and to show that, for series and parallel relations, the polynomial factorises. Our results demonstrate that the Jones polynomial provides a powerful tool for analysing the chirality of molecular knots constructed using circuit topology. The framework presented here can be used to design and engineer a wide range of entangled chain with desired chiral properties, with potential applications in fields such as materials science and nanotechnology.Comment: 8 pages, 5 figures, 3 table

Biology helps to construct weighted scale free networks

In this work we study a simple evolutionary model of bipartite networks which its evolution is based on the duplication of nodes. Using analytical results along with numerical simulation of the model, we show that the above evolutionary model results in weighted scale free networks. Indeed we find that in the one mode picture we have weighted networks with scale free distributions for interesting quantities like the weights, the degrees and the weighted degrees of the nodes and the weights of the edges.Comment: 15 pages, 7 figures, Revte