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

Detection of circular permutations within protein structures using CE-CP

Motivation: Circular permutation is an important type of protein rearrangement. Natural circular permutations have implications for protein function, stability, and evolution. Artificial circular permutations have also been used for protein studies. However, such relationships are difficult to detect for many sequence and structure comparison algorithms and require special consideration. Results: We developed a new algorithm, called Combinatorial Exten-sion for Circular Permutations (CE-CP), which allows the structural comparison of circularly permuted proteins. CE-CP was designed to be user friendly and is integrated into the RCSB Protein Data Bank. It was tested on two collections of circularly permuted proteins. Pai-rwise alignments can be visualized both in a desktop application or on the web using Jmol and exported to other programs in a variety of formats. Availability: The CE-CP algorithm can be accessed through the RCSB website a