Tiing Error Correction & ExperimentsChen Ho-Lin Chen Nov 8 2005 (Chen)

Self-Assembly Self-Assembly is the process by which simple objects autonomously assemble into complexes. Geometry, dynamics, combinatorics are all important Inorganic: Crystals, supramolecules Organic: Proteins, DNA, cells, organisms Goals: Understand self-assembly, design self-assembling systems A key problem in nano-technology, molecular robotics, molecular computation (Chen)

Applications of Self-Assembly Building blocks of nano-machines. DNA computing. Small electrical devices such as FLASH memory. [Black et. Al. ’03] Nanostructures which “steer” light in the same way computer chips steer electrons. [Percec et. Al. ’03] (Chen)

Self-Assembly of DNA [Winfree] (Chen)

Abstract System Model (Chen)

DNA Tiles Glues = sticky ends Tiles = molecules G4 G3 = G1 G2 [Fu and Seeman, ’93] Glues = sticky ends Tiles = molecules (Chen)

abstract Tile Assembly Model: [Rothemund, Winfree, ’2000] Temperature: A positive integer. (Usually 1 or 2) A set of tile types: Each tile is an oriented rectangle with glues on its corners. Each glue has a non-negative strength (0, 1 or 2). An initial assembly (seed). A tile can attach to an assembly iff the combined strength of the “matched glues” is greater or equal than the temperature. x z x y (Chen)

Example: Sierpinski System [Winfree, ’96] 1 1 1 1 T=2 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 (Chen)

DAO-E Sierpinski triangle experiments Paul Rothemund, Nick Papadakis, Erik Winfree, PLoS Biology 2: e424 (2004) 340nm (Chen)

Theoretical Results Efficiently assembling basic shapes with precisely controlled size and pattern. Constructing N X N squares with O(log n/log log n) tiles. [Adleman, Cheng, Goel, Huang, ’01] Perform universal computation by simulating BCA. [Winfree ’99] Assemble arbitrary shape by O( Kolmogorov complexity) tiles with scaling. [Soloveichik, Winfree ’04] (Chen)

Block Cellular Automata f(X, Y) g(X, Y) X Y (Chen)

Simulating BCA T=2 A series of tiles with format: Growth direction f(X,Y) g(X,Y) A series of tiles with format: X Y Growth direction seed (Chen)

Assemble Arbitrary Shapes Replace each tile by a block. Size of block = O(computation time) computation (Chen)

Tile System Design Library of primitives to use in designing nano-scale structures [Adleman, Cheng, Goel, Huang, ’01] Automate the design process [Adleman, Cheng, Goel, Huang, Kempe, Moisset de espanes, Rothemund ’01] (Chen)

Kinetic System Model (Chen)

kinetic Tile Assembly Model: [Winfree, 1998] A tile can attach at any location. The rate of attachment rf = constant. The rate of detachment rr,b = c e-bG (Chen)

Kinetic model => Abstract model We set the temperature and concentration to rr,T+1 << rr,T < rf << rr,T-1 If a tile attaches with strength < T-1, it is likely to fall off very fast. If a tile is held by strength at least T+1, it is unlikely to fall off (Chen)

Error Correction

Robustness Designs Use biological mechanisms in the process. Add extra structure/modification on the process of DNA self-assembly. Add combinatorial structures to tile systems. Use the original erroneous process, but add more structure to the tile system to do error correction.

Using Biological Mechanisms Use strand invasion [Chen, Cheng, Goel, Huang, Moisset de espanes, ’04] Add protecting tiles [Fujibayashi and Murata, ’04] (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Strand Invasion (cont) (Chen)

Strand Invasion (Chen)

Strand Invasion (Chen)

Example T=2 (Chen)

Example T=2 (Chen)

Example T=2 (Chen)

What can go wrong? T=2 (Chen)

What can go wrong? T=2 (Chen)

Why it may not matter: T=2 (Chen)

Why it may not matter: T=2 (Chen)

What can go really wrong? (Chen)

What can go really wrong? (Chen)

What can go really wrong? (Chen)

Safe attachments and invadable systems Unsafe Definition: A tile system is an invadable system iff for all assemblies that can be grown from the initial assembly, all possible attachments are safe. (Chen)

Using Combinatorial Systems Proofreading tiles [Winfree and Bekbolatov, ’02] Snaked tiles [Chen and Goel, ’04] Self healing tiles [Erik Winfree, ’05] (Chen)

Nucleation Error (T=2) Starting from an initial assembly (Chen)

Nucleation Error (T=2) The first tile attaches with strength 1. (usually falls off fast) (Chen)

Nucleation Error (T=2) The second tile attaches and now both tiles are held by strength 2. (Chen)

Nucleation Error (T=2) Error propagates. (Chen)

Modelling Errors Temperature: A positive integer. (Usually 1 or 2) A set of tile types: Each tile is an oriented rectangle with glues on its corners. Each glue has a non-negative strength (0, 1 or 2). An initial assembly (seed). Rules: A tile can attach to an assembly iff the combined strength of the “matched glues” is greater or equal than the temperature. Tiles with combined strength equal to temperature can fall off. Errors: Once a while, there will be two tiles attach at the same time and both are held by strength at least two after the attachment. We call this an “insufficient attachment”. Our goal: minimize the impact of insufficient attachments (Chen)

Snaked Tile System [Chen, Goel, 2004] Replace a tile by G1b X4 X3 G2a X2 G3b G2b G1a G4a G4b G3a Blunt end (inert) Replace a tile by a block of 4 tiles Internal glues are unique (Chen)

Nucleation Error (T=2) Starting from an initial assembly (Chen)

Nucleation Error (T=2) Two tiles attach and both tiles are held by strength 2. (Chen)

Nucleation Error (T=2) No other tiles can attach. Inert edge (Chen)

Generalization G1 G4 G3 G2 (Chen)

Theoretical Analysis The snake tile design can be extended to 2k by 2k blocks. Prevents tile propagation even after k-1 insufficient attachments happen. (Chen)

Why it works? bad ............ : insufficient attachments : erroneous tiles falling off (Chen)

Why it works? bad ............ : happens with rate O(e-G) * rf : erroneous tiles falling off (Chen)

Theoretical Analysis The snake tile design can be extended to 2k by 2k blocks. Prevents tile propagation even after k-1 insufficient attachments happen. When < k insufficient attachments happened locally, all the erroneous tiles are expected to fall off in time poly(k). (Chen)

Why it works? bad ............ : happens with rate O(e-G) * rf : happens with rate 1/poly(k) (Chen)

Why it works? bad ............ bad : happens with rate O(e-kG) if backward rate >> forward rate (Chen)

Theoretical Analysis If we want to assemble a structure with size N, we can use Snaked Tile System with block size k=O(log N). The assembly process is expected to finish within time Õ(N) and be error free with high probability. (Chen)

Experimental Verification Joint work with Ashish Goel Rebecca Schulman Erik Winfree (Chen)

Snaked Tile System [Chen, Goel, 2004] Replace a tile by G3b G4b G1 G2 G3a X3 X2 G4a Replace a tile by a block of 4 tiles Internal glues are unique G1b X4 Blunt end G2b G1a G2a (Chen)

1 3 1c 3c 2c 4c 2 4 2 4 1c 2c 1 2 4c 4 2 4 (Chen)

Tile sets used in experiments Proofreading block Snaked block 4 3 4 3 1 1 2 2 3c 3c 4c 4c 1c 2c 1c 2c (Chen)

[Schulman, Winfree, DNA 10, 2004] Width-4 Zig-Zag Ribbon [Schulman, Winfree, DNA 10, 2004] 6 tile types (Chen)

AFM of Zig-Zag Ribbons (Chen)

ZZ + Snake Tiles 10 tile types Side A Side B (no glue) 1 3 1c 3c 2c 4c (Chen)

Slow nucleation and growth! ZZ + Snake Tiles 1 3 1c 3c 2c 4c 2 4 1 3 1c 3c 2c 4c 2 4 Side A 2 1 3 1c 3c 2c 4c 2 4 4 2 4 2 4 Side B (no glue) 10 tile types (Chen)

Fast nucleation and growth! ZZf + Snake Tiles 2 4 2c 4c 1c 3c 1 3 2 4 3c 3 Side A 4c 1 1 2 4 2c 4c 1c 3c 1 3 3 2c 1c 1 3 1 3 Side B (no glue) 10 tile types (Chen)

Zig-Zag ribbons used in experiments name description Normal Zig-Zag (ZZ) Side A: Glues 2, 4 Side B: blunt Flipped (ZZf) Side A: Glues 1, 3 Double_sided (ZZ_DS) Side B: Glues 1, 3 Flipped + double_sided (ZZ_DSf) Side B: Glues 2, 4 slow fast slow fast fast slow snake tiles (Chen)

Experiment Results (Chen)

ZZf + 100 nM Snaked block (Chen)

Zig-Zag + 100 nM Snaked block (Chen)

Zig-Zag + 100 nM Proofreading (Chen)

ZZf + 100 nM Proofreading (Chen)

ZZ_DS + 10 nM Proofreading (Chen)

ZZ_DS + 10 nM Snaked block

Ratio of chunks on each side Statistics Ratio of chunks on each side Zig-Zag Side A: glues 2, 4 Side B: glues 1, 3 Side A: glues 1, 3 Side B: glues 2, 4 Snaked block 4.7 4.2 Proofreading block 1.1 1.5 (Chen)

Ratio of tiles on each side Statistics Ratio of tiles on each side Zig-Zag Side A: glues 2, 4 Side B: glues 1, 3 Side A: glues 1, 3 Side B: glues 2, 4 Snaked block 4.3 3.9 Proofreading block 1.0 1.2 (Chen)

Self-healing Tile System [Winfree, ’ 05] Goal: When a big portion of the lattice is removed, it should be able to grow back correctly. Method: For each tile in the original system, we create a unique block in the new system. Idea: Use the block to prevent tile from growing backwards. (Chen)

Assumption Use abstract tile assembly model. Requires a fix set of incoming and outgoing edges for each tile in the original system. (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 destroyed (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 (Chen)

Example: Sierpinski System 1 1 1 1 T=2 1 1 1 1 ? ? (Chen)

Example (T=2) Replace a tile by a block of 4 tiles G3 G4 G1 G2 G3 X4 X2 G4 Replace a tile by a block of 4 tiles Internal glues are unique X1 X3 G1 G2 (Chen)

Poly-amino safe The system is save even when several tiles can form a bigger block before attach to the assembly. (Chen)

Example (T=2) Replace a tile by a block of 4 tiles G3 G4 G1 G2 G3 X4 X2 G4 Replace a tile by a block of 4 tiles Internal glues are unique X1 X3 G1 G2 (Chen)

Poly-amino safe The system is save even when several tiles can form a bigger block before attach to the assembly. X1 X3 G1 G2 (Chen)

Poly-amino safe The system is save even when several tiles can form a bigger block before attach to the assembly. G3 X4 X1 X3 G1 G2 (Chen)

Poly-amino safe The system is save even when several tiles can form a bigger block before attach to the assembly. G3 X4 X2 G4 X1 X3 G1 G2 (Chen)

Poly-amino safe The system is save even when several tiles can form a bigger block before attach to the assembly. G3 X4 X2 G4 X1 X3 G1 G2 (Chen)

Poly-amino Safe The system is save even when several tiles can form a bigger block before attach to the assembly. G3 X4 X2 G4 X1 X3 Attach to assembly G1 G2 (Chen)

Not Poly-amino Safe Replace a tile by a block of 4 tiles G3 G4 G1 G2 G3 X4 X2 G4 Replace a tile by a block of 4 tiles Internal glues are unique X1 X3 G1 G2 (Chen)