A Circuit Topology Approach to Categorizing Changes in Biomolecular Structure

Frontiers in Physics, vol. 8

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 toolkit 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.

Authors 4

1. Otto Schullian

   Max Planck Institute of Colloids and Interfaces

   Affiliation as printed

   Max Planck Institute of Colloids and Interfaces, Department Theory and Biosystems, Potsdam, Germany

2. Jaie C. Woodard

   Harvard University

   Affiliation as printed

   Chemistry and Chemical Biology, Harvard University, Cambridge, MA, United States

3. Arash Tirandaz Aachen Faculty of Mathematics and Natural Sciences Leiden Academic Centre for Drug Research

   Leiden University · Institute for Research in Fundamental Sciences

   Affiliation as printed

   Faculty of Mathematics and Natural Sciences, Leiden Academic Centre for Drug Research, Leiden University, Leiden, Netherlands

   School of Biological Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

4. Alireza Mashaghi corresponding Aachen Faculty of Mathematics and Natural Sciences Leiden Academic Centre for Drug Research

   Leiden University

   Affiliation as printed

   Faculty of Mathematics and Natural Sciences, Leiden Academic Centre for Drug Research, Leiden University, Leiden, Netherlands