DNA Origami Words and Rewriting Systems James Garrett, Natasha - - PowerPoint PPT Presentation

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)).

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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.

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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

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DNA Origami Words and Rewriting Systems Preliminaries

The Jones Monoid

The Jones Monoid Jn (Jones (1983)) has generators {h1, . . . , hn−1} and satisfies three classes of relations:

1 hihjhi = hi for |i − j| = 1 2 hihi = hi 3 hihj = hjhi for |i − j| ≥ 2

Generators and relations can be represented graphically.

1 2 3 4 5 h3 in J5

Hwee Kim (USF) Cumberland Conference 2019 5 / 23

DNA Origami Words and Rewriting Systems Preliminaries

The Jones Monoid

The Jones Monoid Jn (Jones (1983)) has generators {h1, . . . , hn−1} and satisfies three classes of relations:

1 hihjhi = hi for |i − j| = 1 2 hihi = hi 3 hihj = hjhi for |i − j| ≥ 2

Generators and relations can be represented graphically.

h1 h2 h1 = h1 h1h2h1 = h1 in J5

Hwee Kim (USF) Cumberland Conference 2019 5 / 23

DNA Origami Words and Rewriting Systems Preliminaries

The Jones Monoid

The Jones Monoid Jn (Jones (1983)) has generators {h1, . . . , hn−1} and satisfies three classes of relations:

1 hihjhi = hi for |i − j| = 1 2 hihi = hi 3 hihj = hjhi for |i − j| ≥ 2

Generators and relations can be represented graphically.

h1 h1 = h1 h1h1 = h1 in J5

Hwee Kim (USF) Cumberland Conference 2019 5 / 23

DNA Origami Words and Rewriting Systems Preliminaries

The Jones Monoid

The Jones Monoid Jn (Jones (1983)) has generators {h1, . . . , hn−1} and satisfies three classes of relations:

1 hihjhi = hi for |i − j| = 1 2 hihi = hi 3 hihj = hjhi for |i − j| ≥ 2

Generators and relations can be represented graphically.

h1 h3 = h3 h1 h1h3 = h3h1 in J5

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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.

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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.

α1 α2 β1 β2 Scaffolds/Staples h1 1 2 1 2 1 2 2 3 2 3 A partial graphical representation of h1 (left), αi and βi (i = 1, 2) (right). For better visibility, staples are shifted right.

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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 Σn = {αi, βi | 1 ≤ i ≤ n − 1}. How do we define the full graphical representation of a symbol?

α1 1 2 3 4

?

“unit” “context” A graphical representation of α1 when n = 4.

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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 Σn = {αi, βi | 1 ≤ i ≤ n − 1}. How do we define the full graphical representation of a symbol?

α1 ∈ Gmax(4) 1 2 3 4 A graphical representation of α1 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 Σn = {αi, βi | 1 ≤ i ≤ n − 1}. How do we define the full graphical representation of a symbol?

α1 ∈ Gmid(4) “virtual” staple 1 2 3 4 A graphical representation of α1 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 Σn = {αi, βi | 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: Gmax(n) and Gmid(n).

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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.

w1 w2 ⇒ w1 w2 ⇒ w1w2

(Staple) Concatenation of w1 and w2.

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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.

α1 α1β2 α1β2α1 = α1β2 ⇒ ⇒ ⇒ α1 β2 α1β2 α1 Concatenation of α1β2 and α1 in Gmid(3).

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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(w1) = G(w2) under a model G, we say w1 ∼ w2 in G and define a rewriting rule w1 ↔ w2. Rewriting rules are different in different models: For example, α1α2α1 ↔ α1 in Gmax(n), but not in Gmid(n).

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Gmax(n)

We may understand structures in Gmax(n) as a cross product of two independent Jones monoids diagrams.

α1 β2 α2 β1 = α1 α2 × β2 β1

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Gmax(n)

Let γ and δ represent arbitrary generators, and γ denote the complementary generator of γ. Then, we have the following set Rmax(n) of rewriting rules (2,3,4 from Jn).

1 (inter-commutation rule) γiγj ↔ γjγi 2 (idempotency rule) γiγi ↔ γi 3 (intra-commutation rule) γiγj ↔ γjγi for |i − j| ≥ 2 4 γiγjγi ↔ γi for |i − j| = 1

Based on Rmax(n), we can define the set Omax(n) of equivalence classes.

Theorem

For all w1, w2 ∈ Σ∗

n, G(w1) = G(w2) under Gmax(n) iff w1 →∗ w2 using

Rmax(n).

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Gmax(n)

Given n, |Jn| is the Catalan number Cn = 1 n + 1

• 2n

n

• , and the maximum

size of an element in Jn is

• n2

4

• (Dolinka and East (2017); Jones (1983)).

Remark

Given n, |Omax(n)| =

• 1

n + 1

• 2n

n

2

, and the maximum size of an irreducible word in Omax(n) is 2

• n2

4

• .

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Gmid(n)

The following three rules still hold for Gmid(n).

1 (Inter-commutation) γiγj ↔ γjγi 2 (Idempotency) γiγi ↔ γi 3 (Intra-commutation) γiγj ↔ γjγi for |i − j| ≥ 2

We cannot directly introduce the rule γiγjγi ↔ γi for |i − j| = 1.

⇒ α1α2α1 α1

• (a)

α1α2α1β2 ⇒ α1β2 (b) α1 α2 α1 β2 α1 α2 α1 α1 β2

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Gmid(n)

The rewriting rule γiγjγi ↔ γi for |i − j| = 1 holds if real staples in both sides occupy the same set of columns. The following rules hold where δ ∈ {α, β} and v is in the finite set Bn

• f words.

4 δjvγiγi−1γi ↔ δjvγi if j = i − 1 or i − 2. 5 δjvγiγi+1γi ↔ δjvγi if j = i + 1 or i + 2. 6 γiγi−1γivδj ↔ γivδj if j = i − 1 or i − 2. 7 γiγi+1γivδj ↔ γivδj if j = i + 1 or i + 2.

We define Rmid(n) to be the set of these seven types of rules.

Theorem

For all w1, w2 ∈ Σ∗

n, G(w1) = G(w2) under Gmid(n) iff w1 →∗ w2 using

Rmid(n).

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

Question: What is |Omid(n)|? (=The number of distinct graphical structures in Gmid(n)) Given a graphical structure G(w), we have the binary b(w) such that the ith bit equals 1 iff the ith staple and the ith scaffold are straight.

1 1 1 1 1 1 1 1 b = 1 1

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

Question: What is |Omid(n)|? (=The number of distinct graphical structures in Gmid(n)) Given a graphical structure G(w), we have the binary b(w) such that the ith bit equals 1 iff the ith staple and the ith scaffold are straight. Each binary string b(w) is uniquely determined with a tuple p(w) = (a1, b1, . . . , ak, bk) where ai (bi) represents the number

• f ith consecutive 0’s (1’s).

1 1 1 1 1 1 1 1 b = 1 1

p = (0, 3, 2, 3, 3, 4)

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

Let D(p) be the number of possible structures G(w) where p(w) = p. Let Tn = {p(w) | |w| = n}. For p = (a1, b1, . . . , ak, bk) ∈ Tn, we have D(p) =

k

• i=1

D(ai, 0) How can we compute D(n, 0)?

1 1 1 1 1 1 1 1 b = 1 1

p = (0, 3, 2, 3, 3, 4) → D(p) = D(0, 0) × D(2, 0) × D(3, 0)

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

Let D(p) be the number of possible structures G(w) where p(w) = p. Let Tn = {p(w) | |w| = n}. From Gmax(n), we know

• p∈Tn

D(p) = |Omax(n)|. Thus, D(n, 0) =

• 1

n + 1

• 2n

n

2

−

• p∈Tn\{(n,0)}

D(p). Boundary conditions: D(0, 0) = 1, D(0, 1) = 1, D(1, 0) = 0 We can recursively calculate D(p) for all p ∈ Tn.

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

For each structure in D(p), assume that we have x(p) distinct structures in Gmid(n). Then |Omid(n)| =

• p∈Tn

[D(p) × x(p)]. Observation: Each block of straight real scaffolds and staples should be adjacent to a block of columns of 0’s.

α1α2α3α2α1 α1 α1α2α1 impossible!! 1 1 1 1 1 1 1 1

A block of straight real scaffolds and staples and a block of columns of 0’s in Gmid(4).

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

Given p = (a1, b1, . . . , ak, bk) ∈ Tn, we can calculate x(p). For each bi, we count the number of possible “virtual” blocks and multiply them.

1 1 1 1 1 1 1 1 b = 1 1

p = (0, 3, 2, 3, 3, 4) → x(p) = 4 × 7 × 5 = 140

4 cases 5 cases 3 + 2 + 1 + 1 = 7 cases

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

Given p = (a1, b1, . . . , ak, bk) ∈ Tn, we can calculate x(p). For each bi, we count the number of possible “virtual” blocks and multiply them. Formally, x(a1, b1, . . . , ak, bk) =

(b1 + 1) if k = 1, (b1 + 1) ·

k−1

• i=2

bi(bi + 1) 2 + 1

• · (bk + 1) if k = 1, a1 = 0,

k−1

• i=1

bi(bi + 1) 2 + 1

• · (bk + 1) if k = 1, a1 > 0.

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DNA Origami Words and Rewriting Systems DNA Origami Rewriting Systems

Rewriting Systems—Counting |Omid(n)|

One exception: When p = (0, n), D(p) × x(p) = n + 1 but there is

• nly one word ǫ—the empty word.

Theorem

|Omid(n)| =

• p∈Tn

[D(p) × x(p)] − n. (|Omid(n)|) = (1, 4, 31, 253, 2247, 21817, 227326, 2499598, 28660639, 339816259 . . .) (new in OEIS)

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DNA Origami Words and Rewriting Systems Conclusions

Conclusions and Future Works

For further reference, the paper will be presented at UCNC 2019 (Garrett et al. (2019)) at June. Future works—possible extensions of basic modules

We’ve only considered rectangular structures. We didn’t consider borderline conditions. We didn’t consider crossings of staples on a scaffold.

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DNA Origami Words and Rewriting Systems Conclusions

Acknowledgments

This work is partially supported by NIH R01GM109459, and by NSF’s CCF-1526485, DMS-1800443 and DMS-1764366.

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DNA Origami Words and Rewriting Systems Conclusions

References I

Dolinka, I. and East, J. (2017). The idempotent-generated subsemigroup

• f the Kauffman monoid. Glasgow Mathematical Journal,

59(3):673–683. Endo, M., Yangyang, Y., and Sugiyama, H. (2013). Dna origami technology for biomaterials applications. Biomaterials Science, 1(4):347–360. Garrett, J., Jonoska, N., Kim, H., and Saito, M. (2019). Dna origami words and rewriting systems. In Proceedings of the 18th International Conference on Unconventional Computation and Natural Computation. accepted. Jones, V. F. R. (1983). Index for subfactors. Inventiones Matheematicae, 72:1–25. Lau, K. W. and FitzGerald, D. G. (2006). Ideal structure of the Kauffman and related monoids. Communications in Algebra, 34(7):2617–2629.

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DNA Origami Words and Rewriting Systems Conclusions

References II

Rothemund, P. W. K. (2006). Folding DNA to create nanoscale shapes and patterns. Nature, 440(7082):297–302.

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