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## Complexities for Generalized Models of Self-Assembly

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**Complexities for Generalized Models of Self-Assembly**Gagan Aggarwal Stanford University Michael H. Goldwasser St. Louis University Ming-Yang Kao Northwestern University Robert T. Schweller Northwestern University Some results were obtained independantly by Cheng, Espanes 2003**Outline**• Importance of DNA Self-Assembly • Synthesis of Nanostructures • DNA Computing • Tile Self-Assembly • DNA Word Design**TILE**G C A T C G C G T A G C**TILE**G C A T C G C G T A G C**Molecular-scale pattern for a RAM memory with demultiplexed**addressing (Winfree, 2003)**DNA Computers**+ Output! Computer Program Input**DNA Computers**+ Output! Computer Program Input Program**DNA Computers**+ Output! Computer Program Input + Input Program**DNA Computers**+ Output! Computer Program Input + Output! Input Program**Outline**• Importance of DNA Self-Assembly • Tile Self-Assembly (Generalized Models) • Tile Complexity • Shape Verification • Error Resistance • DNA Word Design**Tile Model of Self-Assembly**(Rothemund, Winfree STOC 2000) Tile System: t : temperature, positive integer G: glue function T: tileset s: seed tile**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**How a tile system self assembles**G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2 T =**New Models**• Multiple Temperature Model • temperature may go up and down • Flexible Glue Model • Remove the restriction that G(x, y) = 0 for x!=y • Multiple Tile Model • tiles may cluster together before being added • Unique Shape Model • unique shape vs. unique supertile**New Models**• Multiple Temperature Model • temperature may go up and down • Flexible Glue Model • Remove the restriction that G(x, y) = 0 for x!=y • Multiple Tile Model • tiles may cluster together before being added • Unique Shape Model • unique shape vs. unique supertile**New Models**• Multiple Temperature Model • temperature may go up and down • Flexible Glue Model • Remove the restriction that G(x, y) = 0 for x!=y • Multiple Tile Model • tiles may cluster together before being added • Unique Shape Model • unique shape vs. unique supertile**New Models**• Multiple Temperature Model • temperature may go up and down • Flexible Glue Model • Remove the restriction that G(x, y) = 0 for x!=y • Multiple Tile Model • tiles may cluster together before being added • Unique Shape Model • unique shape vs. unique supertile**Focus**• Multiple Temperature Model • Adjust temperature during assembly • Flexible Glue Model • Remove the restriction that G(x, y) = 0 for x!=y Goal: Reduce Tile Complexity**Our Tile Complexity Results**Multiple temperature model: (our paper) k x N rectangles: beats standard model: (our paper) Flexible Glue: N x N squares: (our paper) (Adleman, Cheng, Goel, Huang STOC 2001) beats standard model:**Building k x N Rectangles**k-digit, base N(1/k) counter: k N**Building k x N Rectangles**k-digit, base N(1/k) counter: k N Tile Complexity:**Build a 4 x 256 rectangle:**t = 2 S3 0 S2 0 S1 0 S g g g p C0 C1 C2 C3 S**Build a 4 x 256 rectangle:**t = 2 S3 0 g S2 0 0 1 2 3 0 0 g S1 0 S g g g p C0 C1 C2 C3 0 S3 0 S2 0 0 S1 g g p S C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 g S1 0 S g g g p C0 C1 C2 C3 S3 0 0 S2 0 0 S1 0 0 p S C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 g S1 0 S g g g p C0 C1 C2 C3 S3 0 0 S2 0 0 g g S1 0 0 0 1 S C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 g S1 0 S g g g p C0 C1 C2 C3 S3 0 0 0 0 S2 0 0 0 0 S1 0 0 0 1 p S C1 C2 C3 C0 C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 S g g g p 2 3 C0 C1 C2 C3 S3 0 0 0 0 0 0 S2 0 0 0 0 0 0 S1 0 0 0 1 1 1 p S C1 C2 C3 C0 C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 S3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S1 0 0 0 1 1 1 1 2 2 2 2 3 3 3 p S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 S3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S1 0 0 0 1 1 1 1 2 2 2 2 3 3 3 P S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 S3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 S1 0 0 0 1 1 1 1 2 2 2 2 3 3 3 P S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 S3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 S1 0 0 0 1 1 1 1 2 2 2 2 3 3 3 P R S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 S3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 S1 0 0 0 1 1 1 1 2 2 2 2 3 3 3 P R … S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 S3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 … S1 0 0 0 1 1 1 1 2 2 2 2 3 3 3 P R 0 0 S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2**Build a 4 x 256 rectangle:**t = 2 g g 0 1 0 1 S3 0 p r g S2 0 0 1 2 3 0 0 1 2 g S1 0 p r S P R g g g p 3 0 2 3 p r C0 C1 C2 C3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 P 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 P 3 3 P R 0 0 0 1 1 1 1 2 2 2 2 3 3 3 P C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3**Building k x N Rectangles**k-digit, base N(1/k) counter: k N Tile Complexity:**2-temperature model**t= 4 3 1 3 3**2-temperature model**t = 4 6