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Structures of Monoids Motivated by DNA Origami

Peter Alspaugh, James Garrett, Nataša Jonoska, Masahico Saito

TL;DR

This work defines origami monoids $\mathcal{O}_n$ by doubling the Jones monoid generators to $\alpha_i$ and $\beta_i$ and adding a DNA-inspired rewriting system. It proves $\mathcal{O}_n$ is finite and provides a regular form and a conjectured normal form, while showing a bijection between Green's $\mathscr{D}$-classes of $\mathcal{O}_n$ and those of $\mathcal{J}_n\times\mathcal{J}_n$ via a projection $p$. The approach links the algebraic structure of $\mathcal{O}_n$ to a direct product of Jones monoids and introduces contextual commutation as a key new phenomenon, suggesting a general framework for constructing finite diagrammatic monoids from existing presentations. The results highlight a pathway to analyze larger monoid classes using generator doubling, substitution-type relations, and Green's-class correspondence, with potential extensions to more generator types and other diagrammatic systems.

Abstract

We construct a class of monoids, called origami monoids, motivated by Jones monoids and by strand organization in DNA origami structures. Two types of basic building blocks of DNA origami closely associated with the graphical representation of Jones monoids are identified and are taken as generators for the origami monoid. Motivated by plausible modifications of the DNA origami structures and the relations of the well studied Jones monoids, we then identify a set of relations that characterize the origami monoid. These relations expand the relations of the Jones monoids and include a new set of relations called contextual commutation. With contextual commutation, certain generators commute only when found within a given context. We prove that the origami monoids are finite and propose a normal form representation of their elements. We establish a correspondence between the Green's classes of the origami monoid and the Green's classes of a direct product of Jones monoids.

Structures of Monoids Motivated by DNA Origami

TL;DR

This work defines origami monoids by doubling the Jones monoid generators to and and adding a DNA-inspired rewriting system. It proves is finite and provides a regular form and a conjectured normal form, while showing a bijection between Green's -classes of and those of via a projection . The approach links the algebraic structure of to a direct product of Jones monoids and introduces contextual commutation as a key new phenomenon, suggesting a general framework for constructing finite diagrammatic monoids from existing presentations. The results highlight a pathway to analyze larger monoid classes using generator doubling, substitution-type relations, and Green's-class correspondence, with potential extensions to more generator types and other diagrammatic systems.

Abstract

We construct a class of monoids, called origami monoids, motivated by Jones monoids and by strand organization in DNA origami structures. Two types of basic building blocks of DNA origami closely associated with the graphical representation of Jones monoids are identified and are taken as generators for the origami monoid. Motivated by plausible modifications of the DNA origami structures and the relations of the well studied Jones monoids, we then identify a set of relations that characterize the origami monoid. These relations expand the relations of the Jones monoids and include a new set of relations called contextual commutation. With contextual commutation, certain generators commute only when found within a given context. We prove that the origami monoids are finite and propose a normal form representation of their elements. We establish a correspondence between the Green's classes of the origami monoid and the Green's classes of a direct product of Jones monoids.
Paper Structure (14 sections, 17 theorems, 16 equations, 18 figures, 2 tables)

This paper contains 14 sections, 17 theorems, 16 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jones monoids
  4. String rewriting systems
  5. Monoids and Green's relations
  6. Origami monoid $\mathcal{O}_n$
  7. Generators
  8. Concatenation as a monoid operation
  9. Relations in $\mathcal{O}_n$
  10. Defining relations of $\mathcal{O}_n$
  11. Additional rewriting rules derived by substitution
  12. Towards normal forms
  13. Green's classes
  14. Concluding remarks

Key Result

Lemma 1

Let $i,j \in \{1,2,\dots ,n-1\}$ such that $|i-j|=1$ and let $\gamma \in \{\alpha, \beta\}$. Then we have the following:

Figures (18)

  • Figure 1: A schematics of DNA origami structure with scaffold in black and staples in color (edited from rothemund-folding-2006).
  • Figure 2: (A) The generators $h_i$ of $\mathcal{J}_n$ and relations (B,C,D) of the Jones monoid $\mathcal{J}_n$temperley.
  • Figure 3: $\alpha_i$
  • Figure 4: $\beta_i$
  • Figure 5: $\alpha_4$ in the context of a 6-fold stranded structure
  • ...and 13 more figures

Theorems & Definitions (36)

  • Example 1
  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Corollary 4
  • ...and 26 more