Topology of RNA-RNA interaction structures

TL;DR

This paper explores the topological classification of RNA-RNA interaction structures, establishing finiteness results for fixed genus and developing a formal grammar for genus zero structures that enables efficient computation of thermodynamic properties.

Contribution

It introduces the concept of irreducible shadows for classifying RNA interaction structures and provides a computational framework for genus zero structures with practical algorithms.

Findings

Finitely many irreducible shadows exist for fixed genus.

Genus zero interaction structures can be described by a multiple context-free grammar.

The proposed grammar computes minimum free energy, partition function, and base pairing probabilities in polynomial time.

Abstract

The topological filtration of interacting RNA complexes is studied and the role is analyzed of certain diagrams called irreducible shadows, which form suitable building blocks for more general structures. We prove that for two interacting RNAs, called interaction structures, there exist for fixed genus only finitely many irreducible shadows. This implies that for fixed genus there are only finitely many classes of interaction structures. In particular the simplest case of genus zero already provides the formalism for certain types of structures that occur in nature and are not covered by other filtrations. This case of genus zero interaction structures is already of practical interest, is studied here in detail and found to be expressed by a multiple context-free grammar extending the usual one for RNA secondary structures. We show that in $O(n^6)$ time and $O(n^4)$ space complexity, this grammar for genus zero interaction structures provides not only minimum free energy solutions but also the complete partition function and base pairing probabilities.