A Circuit Topology Approach to Categorizing Changes in Biomolecular Structure

Abstract

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.

Protein structural features predict responsiveness to pharmacological chaperone treatment for three lysosomal storage disorders

“…The interval of a contact is defined as the span of sequence between the two contacting residues, along the protein chain. Two contacts may be in three types of relations: series, parallel, or cross, as illustrated in Figure 1 and described in previous publications (30)(31)(32). In particular, the parallel relation, relevant to this study, occurs when one interval is contained entirely within the other.…”

“…Number of residues in contact with the mutated residue was calculated for each residue type by considering the number of residues forming greater than or equal to five atom-atom pairwise contacts of any type within 4.5 Angstroms. Circuit topology relations were calculated using existing code and methods (30). For each residue, the number of relations of a given type with respect to any contact formed by the mutated residue was considered as "local" circuit topology for that residue.…”

Generalized Circuit Topology of Folded Linear Chains

“…Circuit topology can be extended to three-point contacts, which though is rarely needed in practice, especially while dealing with biomolecules ( Mashaghi et al., 2014 ) ( Schullian et al., 2020 ). Three-point contacts are considered as an extreme case when sites of two-point contacts are merging together, and hence their sequence is not clear, A(AB)B versus A(BA)B , and (AB)BA versus (BA)BA .…”

“…If two chains merge, their strings merge in the same manner. These and other changes in a chain structure were recently considered in Schullian et al. (2020) by using a connectivity matrix S that shows which contact sites are connected.…”

“…For binary contacts, those arrangements are parallel (P), series (S), and cross (X). The approach allows for topology classification and for extracting topological information from a folded chain, which is suggested to be useful in understanding folding pathways and kinetics as well as in predicting functional and evolutionary relatedness of biomolecules ( Heidari et al., 2017 ) ( Heidari et al., 2019 ) ( Mugler et al., 2014 ) ( Nikoofard and Mashaghi, 2016 ) ( Nikoofard and Mashaghi, 2018 ) ( Satarifard et al., 2017 ) ( Schullian et al., 2020 ). The approach is, however, limited as it does not include chain entanglement, a topic that is addressed by knot theory ( Adams, 2004 ).…”