Contents lists available at ScienceDirect Reactive and Functional Polymers journal homepage: www.elsevier.com/locate/react The APS-bracket – A topological tool to classify lasso proteins, RNAs and other tadpole-like structures Pawel Dabrowski-Tumanski a,b , Joanna I. Sulkowska a,b, ⁎ a Faculty of Chemistry, University of Warsaw, Pasteura 1, Warsaw 02-093, Poland b Centre of New Technologies, University of Warsaw, Banacha 2c, Warsaw 02-097, Poland ARTICLE INFO Keywords: Lasso proteins Tadpole Topology Kauffman bracket Knot invariants ABSTRACT The topology of biopolymers plays a crucial role in their function and properties. The standard knot-theory techniques allow one to distinguish the topology of linear polymers, however, they do not take into account the intrinsic and external interactions in the biopolymer chain, such as hydrogen or disulfide bonds. Here we in- troduce the APS-bracket – a topological tool developed to analyze the topology of biopolymers with intrachain interactions. We list the basic properties of the bracket and calculate its value for structures inspired by complex lasso proteins. We also introduce a set of basis structures and define the input of each separate basis structure. Finally, we discuss possible applications and generalizations of the bracket. 1. Introduction The reactivity and function of polymers depend both on their local and global properties. One of the global properties is the topology of the polymer, which alters, for example, the stability [1–6] or the hydro- dynamical properties [7,8]. In the case of non-branched, closed poly- mers, their topology may be effectively described using the tools from knot theory. In the case of open chain polymers such as proteins, the topological description requires an additional step of chain closure. Some methods for reliable chain closure have been developed [9,10] that allow one to define the topology of open-chain polymers as well. In particular, it was shown that a sufficiently long polymer is most probably knotted [11,12]. Indeed, knots have been identified in DNA [13,14] (but not in RNA [15,16]) and proteins [17–22], despite possible problems with protein folding [23,24]. Nevertheless, the real polymers are rarely non-interacting. On the contrary, the building block of DNA/ RNA (nucleotide bases) and proteins (amino acids) form a dense net of interactions via hydrophobic, hydrogen or covalent bonds. These in- teractions are usually much weaker than the covalent bond in the polymer backbone. However, they modulate the shape, the topology, and therefore the function and properties of biopolymers. Therefore, it is desirable to include such interactions in the description of the to- pology of polymers. The aim of this work is to introduce a topological tool which we call the APS-bracket. It is built as an extension of the Kauffman bracket used widely to analyze the knotted topology [25]. As our goal is to use this tool to classify the existing biopolymers, first we briefly review the current knowledge about complex topology in proteins and DNA. This motivates the properties of the bracket, which we build in the next section. With the bracket constructed, we investigate its further prop- erties. In particular, we introduce the basis of structures with loops, the projection onto this basis set, and relate the bracket to Kauffman's Simplified RNA Polynomial [26]. We also calculate the bracket for the exemplary structures, including graphs corresponding to some complex lasso proteins present in PDB. Finally, we analyze possible applications and extensions of our bracket. 2. Results 2.1. Topology in RNA and DNA In addition to the links arising naturally in duplication of circular DNA [13,27,28], the topology of nucleic acids features other complex structures. One can distinguish, for example, between the complicated arrangements of hydrogen bonds in an RNA strand forming stem-loops, pseudoknots and other structures (Fig. 1A). To build the mathematical classification of the RNA structures, we describe them as spatial graphs in which the vertices represent the nucleotide bases. As the bases in- teract usually in pairs in the Watson-Crick mechanism, such graphs feature only two- or trivalent vertices (we assume that the chain is closed, as otherwise it may be artificially closed using standard tech- niques [10,17]), where the valency depends on whether the https://doi.org/10.1016/j.reactfunctpolym.2018.09.005 Received 5 April 2018; Received in revised form 20 August 2018; Accepted 7 September 2018 ⁎ Corresponding author at: Faculty of Chemistry, University of Warsaw, Pasteura 1, Warsaw 02-093, Poland. E-mail address: jsulkowska@chem.uw.edu.pl (J.I. Sulkowska). Reactive and Functional Polymers 132 (2018) 19–25 Available online 11 September 2018 1381-5148/ © 2018 Elsevier B.V. All rights reserved. T