Algebraic Systems for DNA Origami Motivated from Temperley-Lieb Algebras

TL;DR

This paper introduces an algebraic framework for DNA origami structures using monoids inspired by Temperley-Lieb algebras, providing a new mathematical perspective on their composition and relations.

Contribution

It defines the origami monoid with generators and relations based on DNA building blocks, connecting it to Temperley-Lieb and Jones algebras for the first time.

Findings

Identification of two basic DNA origami building blocks.

Definition of the origami monoid with specific relations.

Analysis of Green's relations and connections to Jones monoids.

Abstract

We initiate an algebraic approach to study DNA origami structures by associating an element from a monoid to each structure. We identify two types of basic building blocks and describe an DNA origami structure with their composition. These building blocks are taken as generators of a monoid, called origami monoid, and, motivated by the well studied Temperley-Lieb algebras, we identify a set of relations that characterize the origami monoid. We also present several observations about the Green's relations for the origami monoid and study the relations to a cross product of Jones monoids that is a morphic image of an origami monoid.