DNA Origami Words and Rewriting Systems DNA Origami Words and Rewriting Systems James Garrett, Natasha Jonoska, Hwee Kim ∗ and Masahico Saito Department of Mathematics and Statistics, University of South Florida The 31st Cumberland Conference on Combinatorics, Graph Theory and Computing Hwee Kim (USF) Cumberland Conference 2019 1 / 23
DNA Origami Words and Rewriting Systems Introduction Motivation—DNA Origami DNA Origami engineers DNA strands to self-interact and construct a microscopic structure autonomously. DNA origami structure construction (Endo et al. (2013)). Hwee Kim (USF) Cumberland Conference 2019 2 / 23
DNA Origami Words and Rewriting Systems Introduction Motivation—DNA Origami Scaffold (a long single strand) + Staples (short strands) Can we represent a structure by composition of basic modules? A schematic representation of a DNA origami structure (Rothemund (2006)). The scaffold is a black line and staples are colored lines with arrows. Hwee Kim (USF) Cumberland Conference 2019 3 / 23
DNA Origami Words and Rewriting Systems Introduction Motivation—DNA Origami Scaffold (a long single strand) + Staples (short strands) Can we represent a structure by composition of basic modules? Basic modules: Type α and type β . Hwee Kim (USF) Cumberland Conference 2019 3 / 23
DNA Origami Words and Rewriting Systems Introduction Our Results Define basic modules for a DNA origami structure, and associate symbols to modules. Define concatenation of symbols by merging origami structures. Define rewriting rules and systems based on equivalence of structures. Analyze properties of DNA origami rewriting systems: The number of distinct equivalence classes The size of a maximum word within each class A polytime algorithm that finds the shortest length word within the class Hwee Kim (USF) Cumberland Conference 2019 4 / 23
DNA Origami Words and Rewriting Systems Preliminaries The Jones Monoid The Jones Monoid J n (Jones (1983)) has generators { h 1 , . . . , h n − 1 } and satisfies three classes of relations: 1 h i h j h i = h i for | i − j | = 1 2 h i h i = h i 3 h i h j = h j h i for | i − j | ≥ 2 Generators and relations can be represented graphically. 1 2 3 4 5 h 3 in J 5 Hwee Kim (USF) Cumberland Conference 2019 5 / 23
DNA Origami Words and Rewriting Systems Preliminaries The Jones Monoid The Jones Monoid J n (Jones (1983)) has generators { h 1 , . . . , h n − 1 } and satisfies three classes of relations: 1 h i h j h i = h i for | i − j | = 1 2 h i h i = h i 3 h i h j = h j h i for | i − j | ≥ 2 Generators and relations can be represented graphically. h 1 h 2 = h 1 h 1 h 1 h 2 h 1 = h 1 in J 5 Hwee Kim (USF) Cumberland Conference 2019 5 / 23
DNA Origami Words and Rewriting Systems Preliminaries The Jones Monoid The Jones Monoid J n (Jones (1983)) has generators { h 1 , . . . , h n − 1 } and satisfies three classes of relations: 1 h i h j h i = h i for | i − j | = 1 2 h i h i = h i 3 h i h j = h j h i for | i − j | ≥ 2 Generators and relations can be represented graphically. h 1 = h 1 h 1 h 1 h 1 = h 1 in J 5 Hwee Kim (USF) Cumberland Conference 2019 5 / 23
DNA Origami Words and Rewriting Systems Preliminaries The Jones Monoid The Jones Monoid J n (Jones (1983)) has generators { h 1 , . . . , h n − 1 } and satisfies three classes of relations: 1 h i h j h i = h i for | i − j | = 1 2 h i h i = h i 3 h i h j = h j h i for | i − j | ≥ 2 Generators and relations can be represented graphically. h 3 h 1 = h 3 h 1 h 1 h 3 = h 3 h 1 in J 5 Hwee Kim (USF) Cumberland Conference 2019 5 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words DNA Origami Words Two differences of DNA origami structures from the Jones monoid: 1 DNA origami has two different types of lines: scaffolds and staples. 2 DNA strands are oriented. Basic modules: Type α and type β . Hwee Kim (USF) Cumberland Conference 2019 6 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words DNA Origami Words Two differences of DNA origami structures from the Jones monoid: 1 DNA origami has two different types of lines: scaffolds and staples. 2 DNA strands are oriented. Scaffolds/Staples 1 2 1 2 2 3 1 2 2 3 h 1 α 1 α 2 β 1 β 2 A partial graphical representation of h 1 (left), α i and β i ( i = 1 , 2) (right). For better visibility, staples are shifted right. Hwee Kim (USF) Cumberland Conference 2019 6 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words DNA Origami Words Given n as the width of the structure, we use the set of symbols Σ n = { α i , β i | 1 ≤ i ≤ n − 1 } . How do we define the full graphical representation of a symbol? ? 1 2 3 4 “unit” “context” α 1 A graphical representation of α 1 when n = 4. Hwee Kim (USF) Cumberland Conference 2019 7 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words DNA Origami Words Given n as the width of the structure, we use the set of symbols Σ n = { α i , β i | 1 ≤ i ≤ n − 1 } . How do we define the full graphical representation of a symbol? 1 2 3 4 α 1 ∈ G max (4) A graphical representation of α 1 when n = 4. Hwee Kim (USF) Cumberland Conference 2019 7 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words DNA Origami Words Given n as the width of the structure, we use the set of symbols Σ n = { α i , β i | 1 ≤ i ≤ n − 1 } . How do we define the full graphical representation of a symbol? “virtual” staple 1 2 3 4 α 1 ∈ G mid (4) A graphical representation of α 1 when n = 4. Virtual staples represent missing staples, and used for simpler definition of concatenation. Hwee Kim (USF) Cumberland Conference 2019 7 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words DNA Origami Words Given n as the width of the structure, we use the set of symbols Σ n = { α i , β i | 1 ≤ i ≤ n − 1 } . How do we define the full graphical representation of a symbol? According to different “contexts” for generators, we consider two different types of sets of graphical structures: G max ( n ) and G mid ( n ) . Hwee Kim (USF) Cumberland Conference 2019 7 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words Concatenation of Words Concatenation is defined similarly as in the Jones monoid diagrams, with the following additions: Scaffolds and staples are connected independently. If a virtual staple meets a real one, it becomes real—in other words, real staples extend. w 1 w 1 ⇒ ⇒ w 1 w 2 w 2 w 2 (Staple) Concatenation of w 1 and w 2 . Hwee Kim (USF) Cumberland Conference 2019 8 / 23
DNA Origami Words and Rewriting Systems DNA Origami Words Concatenation of Words Concatenation is defined similarly as in the Jones monoid diagrams, with the following additions: Scaffolds and staples are connected independently. If a virtual staple meets a real one, it becomes real—in other words, real staples extend. α 1 ⇒ β 2 α 1 α 1 β 2 α 1 β 2 ⇒ ⇒ α 1 α 1 β 2 α 1 = α 1 β 2 Concatenation of α 1 β 2 and α 1 in G mid (3) . Hwee Kim (USF) Cumberland Conference 2019 8 / 23
DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems Rewriting Systems Given a word w , we can incrementally construct the graphical structure G ( w ), which consists of pairs of top and bottom points. If G ( w 1 ) = G ( w 2 ) under a model G , we say w 1 ∼ w 2 in G and define a rewriting rule w 1 ↔ w 2 . Rewriting rules are different in different models: For example, α 1 α 2 α 1 ↔ α 1 in G max ( n ) , but not in G mid ( n ) . Hwee Kim (USF) Cumberland Conference 2019 9 / 23
DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems Rewriting Systems— G max ( n ) We may understand structures in G max ( n ) as a cross product of two independent Jones monoids diagrams. α 1 α 1 β 2 β 2 = × α 2 α 2 β 1 β 1 Hwee Kim (USF) Cumberland Conference 2019 10 / 23
DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems Rewriting Systems— G max ( n ) Let γ and δ represent arbitrary generators, and γ denote the complementary generator of γ . Then, we have the following set R max ( n ) of rewriting rules (2,3,4 from J n ). 1 (inter-commutation rule) γ i γ j ↔ γ j γ i 2 (idempotency rule) γ i γ i ↔ γ i 3 (intra-commutation rule) γ i γ j ↔ γ j γ i for | i − j | ≥ 2 4 γ i γ j γ i ↔ γ i for | i − j | = 1 Based on R max ( n ) , we can define the set O max ( n ) of equivalence classes. Theorem For all w 1 , w 2 ∈ Σ ∗ n , G ( w 1 ) = G ( w 2 ) under G max ( n ) iff w 1 → ∗ w 2 using R max ( n ) . Hwee Kim (USF) Cumberland Conference 2019 11 / 23
DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems Rewriting Systems— G max ( n ) � � 1 2 n Given n , |J n | is the Catalan number C n = , and the maximum n + 1 n � � n 2 size of an element in J n is (Dolinka and East (2017); Jones (1983)). 4 Remark �� 2 � � 1 2 n Given n , |O max ( n ) | = , and the maximum size of an n + 1 n � n 2 � irreducible word in O max ( n ) is 2 . 4 Hwee Kim (USF) Cumberland Conference 2019 12 / 23
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