RNA topology: From simple loops to complex circuits - A review

RNA molecules fold into complex three-dimensional structures that determine their function. A wide range of mathematical frameworks, such as chord diagrams, fatgraphs, and context-free grammars, have been used to represent these structures; however, these models have largely been developed from mathematical motivations rather than from considerations of the physical principles that govern the folding, organization, and interactions of RNA molecules. In this perspective review, we reinterpret these mathematical models through the lens of circuit topology (CT), a physics-inspired approach originally developed to characterize interaction patterns in linear polymers, relate them to folding kinetics, and enable molecular design and engineering. Using CT, RNA secondary structures, including pseudoknots, can be represented compactly, revealing a direct mapping between traditional RNA models and the underlying topological arrangement of interactions. This unified view highlights shared structural principles between RNA and other polymer systems and provides a common basis for exchanging methods across different areas of molecular science. Finally, we outline how the CT framework can be extended to analyze higher-order motifs such as RNA-RNA interactions, base triples, and aspects of folding kinetics. By providing a common language for analyzing the organization and function of folded biomolecular chains, CT bridges mathematical abstraction and molecular reality.