1 January 18, 2010 Shape Replication through Self-Assembly and Rnase Enzymes Zachary AbelHarvard University Nadia BenbernouMassachusetts Institute of Technology Mirela DamianVillanova University Erik D. DemaineMassachusetts Institute of Technology Martin DemaineMassachusetts Institute of Technology Robin FlatlandSiena College Skott D. KominersHarvard University Robert SchwellerUniversity of Texas Pan American Read: Replicate:

2 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

3 Tile Assembly Model (Rothemund, Winfree, Adleman) T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 Tile Set: Glue Function: Temperature: x ed cba

4 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

5 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

6 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

7 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

8 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

9 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

10 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

11 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

12 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

13 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

14 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba x abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

15 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 abc d e x x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

16 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx Tile Assembly Model (Rothemund, Winfree, Adleman)

17 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx x Tile Assembly Model (Rothemund, Winfree, Adleman)

18 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx xx (Basic)Tile Assembly Model (Rothemund, Winfree, Adleman)

19 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

20 RNA enzyme Self-Assembly (suggested by Rothemund, Winfree 2000) RNA tile types DNA tile types RNA assembly model: Assembly occurs over a number of stages. At each stage you may: 1) Add a new collection of tile types - Allow for further growth - All added types have infinite count 2) Add an Rnase enzyme - Dissolve all RNA tile types - May break apart assemblies All tile types are of either DNA or RNA makeup:

21 RNA enzyme Self-Assembly Stage 1:

22 RNA enzyme Self-Assembly Stage 1:

23 RNA enzyme Self-Assembly Stage 1: Stage 2:

24 RNA enzyme Self-Assembly Stage 1: Stage 2:

25 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme

26 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme

27 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme

28 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

29 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

30 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

31 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

32 RNA enzyme Self-Assembly Metrics for efficiency: Tile complexity: total number of distinct tile types used in the system. Stage complexity: total number of distinct stages used.

33 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

Shape Replication Problem Design an assembly system (algorithm) that will replicate a large number of copies given a single copy of a pre-assembled input shape. Precise Yield: Replicate exactly n copies for a given n Infinite Yield: Replicate infinite copies -in practice, the number of copies should only be limited by the volume of particles available.

35 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

Precise Yield: rectangles

aaaa a a a a a a aaaa a a a a a a

nnnn e e e e e e w w w w w w ssss

nnnn e e e e e e w w w w w w ssss n w xx y y

nnnn e e e e e e w w w w w w ssss n w

nnnn e e e e e e w w w w w w ssss n w n e s ws e

nnnn e e e e e e w w w w w w ssss n w n e s w s e

n nn n e e e e e e w w w w w w ssss a w a a a

nn e e w w ss Step 1: Coat shape with layer of RNA

Precise Yield: rectangles nn e e w w ss Step 2: Coat shape with layer of DNA Step 1: Coat shape with layer of RNA

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 6: Add enzyme. Step 1: Coat shape with layer of RNA.

Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 6: Add enzyme. Step 1: Coat shape with layer of RNA.

Precise Yield: General Shapes

Precise Yield: rectangles

Tile typesO(1) StagesO(log n) Precise Yield: n copies Precise Yield: rectangles Tile typesO(log n) StagesO(1) Precise Yield: n copies

59 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

Infinite Yield: Rectangles nnnn e e e e e e w w w w w w ssss

nnnn e e e e e e w w w w w w ssss s w a

a s b s b s a b ss a s Stair step tiles: x x

Infinite Yield: Rectangles a s b s b s a b ss Stair step tiles: x x b x

Infinite Yield: Rectangles a s b s b s a b ss Stair step tiles: x x a b

Infinite Yield: Rectangles a s b s b s a b s s Stair step tiles: x x a b

Infinite Yield: Rectangles a s b s b s a b s Stair step tiles: x x b a b

Infinite Yield: Rectangles a s b s b s a b s Stair step tiles: x x b a b

Infinite Yield: Rectangles a s b s b s a b s Stair step tiles: x x b a b

Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x b b b a

Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x

Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x

Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x …

Infinite Yield: Rectangles Tile typesO(1) StagesO(1) Infinite Yield: Rectangles

Infinite Yield: General Shapes

Step 1: Coat with RNA

Infinite Yield: General Shapes Step 2: Create rectangular DNA encasing

Infinite Yield: Binary counter tool cccccccccccccccccccccccc

1 ccccccccccccccccccccccc m nn nn cc nc 1 m m x x Binary counter tiles: c 0 1 mm n

1 Infinite Yield: Binary counter tool 1 cccccccccccccccccccccc 0 0 m nn nn cc nc 1 m m x x Binary counter tiles: c 0 1 mm n

1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccccc 0 m 1 1 n nn nn cc nc 1 m m x x Binary counter tiles: c 0 1 mm n

1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccccc nn nn cc nc 1 m m x x Binary counter tiles: c 0 0 m mm n 1

1 Infinite Yield: Binary counter tool 1 cccccccccccccccccccc nn nn cc nc 1 m m x x Binary counter tiles: c mm n c m

1 Infinite Yield: Binary counter tool 1 cccccccccccccccccccc nn nn cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n

1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccc nn nn cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n n

1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccc nn nn cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n n 0 0

1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccc nn nn cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n

1 Infinite Yield: Binary counter tool nn nn cc nc 1 m m x x Binary counter tiles: c mm n

Infinite Yield: General Shapes … …

… …

Step 3: Label each face with unique binary code

Infinite Yield: General Shapes Step 4: Enzyme.

Infinite Yield: General Shapes … Step 5: Infinitely replicate all labeled rectangles

Infinite Yield: General Shapes

Reassembly?

Infinite Yield: General Shapes Reassembly?

Infinite Yield: General Shapes Reassembly?

Infinite Yield: General Shapes Reassembly?

Infinite Yield: General Shapes Step 6: Reassemble, fill in frame, break out copies with enzyme.

Infinite Yield: General Shapes Tile typesO(1) StagesO(1) Infinite Yield: Vertically convex

Infinite Yield: Non-vertically convex shapes Grow counter along surface of shape

Grow counter along surface of shape Start end Infinite Yield: Non-vertically convex shapes

Grow counter along surface of shape Break apart with enzyme Infinite Yield: Non-vertically convex shapes

Grow counter along surface of shape Break apart with enzyme Replicate Infinite Yield: Non-vertically convex shapes

Grow counter along surface of shape Break apart with enzyme Replicate Reassemble Infinite Yield: Non-vertically convex shapes

Grow counter along surface of shape Break apart with enzyme Replicate Reassemble Infinite Yield: Non-vertically convex shapes

107 Tile typesO(1) StagesO(1) Infinite Yield: Infinite Yield: General Shapes

Future Work Replicate and improve -Hybrid algorithms for replication and modification Extension to 3D -Planarity/spacial constraint Replication of internal pattern Staged enzyme model for assembly from scratch - Seems to be very powerful for this Temperature changes to perform replication

109 January 18, 2010 Thank you. Questions? Zachary AbelHarvard University Nadia BenbernouMassachusetts Institute of Technology Mirela DamianVillanova University Erik D. DemaineMassachusetts Institute of Technology Martin DemaineMassachusetts Institute of Technology Robin FlatlandSiena College Skott D. KominersHarvard University Robert SchwellerUniversity of Texas Pan American Read: Replicate: