symmetry S S Article Coloring Invariant for Topological Circuits in Folded Linear Chains Jose Ceniceros 1 , Mohamed Elhamdadi 2 and Alireza Mashaghi 3, *   Citation: Ceniceros, J.; Elhamdadi, M.; Mashaghi, A. Coloring Invariant for Topological Circuits in Folded Linear Chains. Symmetry 2021, 13, 919. https://doi.org/10.3390/ sym13060919 Academic Editors: Erica Flapan and Helen Wong Received: 18 April 2021 Accepted: 18 May 2021 Published: 21 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Mathematics and Statistics Department, Hamilton College, Clinton, NY 13323, USA; jcenicer@hamilton.edu 2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA; emohamed@usf.edu 3 Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, 2333 CC Leiden, The Netherlands * Correspondence: a.mashaghi.tabari@lacdr.leidenuniv.nl Abstract: Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a folded linear chain, such as a protein molecule or the genome. Theses linear biomolecular chains often fold into complex 3D architectures with critical entanglements and local or global structural symmetries stabilised by formation of intrachain contacts. Here, we adapt and apply the algebraic structure of quandles to classify and distinguish chain topologies within the framework of circuit topology. We systematically study the basic circuit topology motifs and define quandle/bondle coloring for them. Next, we explore the implications of circuit topology operations that enable building complex topologies from basic motifs for the quandle coloring approach. Keywords: circuit topology; folded linear chains; quandle; bondle; coloring invariant 1. Introduction Folded linear molecular chains typically fold via formation of intra-chain bonds that fix chain entanglement; examples include base–base interactions in nucleic acids and residue–residue interactions in proteins [1–3]. During the folding process, interactions form progressively until the chain reaches its final topology. Formation of an interaction may facilitate or hinder formation of subsequent interactions depending on the topological relations between them. Functionally critical entanglements and local or global structural symmetries are then stabilised by the formation of intrachain contacts. Motivated by the physics of folding, circuit topology has been developed to categorize the arrangement of intra-chain interactions (so-called contacts) and chain crossings in a folded linear chain [4]. While being mathematically rigorous, the topological picture is intuitive and simple and provides quantitative measures that can be readily used to relate topology to other proper- ties of folded chain systems [5]. A growing number of studies have recently been conducted in which implications of circuit topology for folding and unfolding dynamics of polymer chains have been investigated [5]. The relation between circuit topology and the well-established knot theory is being studied [6]. While circuit topology complements knot theory by addressing interactions, tools developed by knot theorists might prove useful in strengthening circuit topology. Various coloring schemes have been developed by knot theorists that might be extended to classify topological circuits. Recently, a quandle coloring approach has been developed for protein analysis, which can in principle be adapted to resolve the space of possible circuit topologies [7,8]. The aim of this article is to use algebraic structures of quandles to study circuit topology. Using knot theory tools, we construct an invariant which we use to classify and distinguish chain topologies within the framework of circuit topology. The article is organized as follows. In Section 2, we review the basics of circuit topology. Section 3 reviews the basics of quandles, singquandle, and bondles with examples. In Symmetry 2021, 13, 919. https://doi.org/10.3390/sym13060919 https://www.mdpi.com/journal/symmetry