Watson-Crick pairing, the Heisenberg group and Milnor invariants

TL;DR

This paper explores the use of Milnor invariants, specifically the Heisenberg invariant, to analyze RNA secondary structures, providing bounds on unpaired bases and predicting allosteric structures.

Contribution

It introduces the Heisenberg invariant as a novel topological tool for RNA structure analysis, linking knot theory with biological structure prediction.

Findings

Heisenberg invariant bounds the number of unpaired bases.

Large Heisenberg invariant indicates presence of allosteric structures.

The invariant can be interpreted via the Heisenberg group and lattice paths.

Abstract

We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict \emph{allosteric structures} for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson-Crick pairs found.