Pattern overlap implies runaway growth in hierarchical tile systems - PowerPoint PPT Presentation

Self-assembly Main results Conclusions Pattern overlap implies runaway growth in hierarchical tile systems Ho-Lin Chen 1 David Doty 2 nuch 3 , 4 J´ an Maˇ Arash Rafiey 4 , 5 Ladislav Stacho 4 1 National Taiwan University, Taiwan 2 California Institute of Technology, USA 3 University of British Columbia, Canada 4 Simon Fraser University, Canada 5 Indiana State University, USA SoCG 2015 SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Molecular self-assembly Engineering goal SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

DNA DNA tile self-assembly DNA 7

DNA tile self-assembly DNA tile self-assembly DNA tile DNA tile (branched junction, Seeman, Journal of Theoretical Biology 1982) sticky end 8

DNA tile self-assembly DNA tile self-assembly DNA tile DNA tile (branched junction, Seeman, Journal of Theoretical Biology 1982) sticky end 9

DNA tile self-assembly DNA tile self-assembly Many copies of DNA tile in solution ... ... ... ... ... 10

DNA tile self-assembly DNA tile self-assembly Many copies of DNA tile in solution ... ... Liu, Zhong, Wang, Seeman, Angewandte Chemie 2011 ... ... ... 11

DNA tile self-assembly DNA tile self-assembly Using more types of tiles (i.e., different sticky ends)... Wei, Dai, Yin, Nature 2012 12

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) 15

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) ● tile type = unit square 16

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) ● tile type = unit square ● each side has a glue , with a label and a strength (0,1,2) strength 0 strength 1 strength 2 17

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) ● tile type = unit square ● each side has a glue , with a label and a strength (0,1,2) ● if tiles with matching glues touch, they're strength 0 attracted with the glue's strength strength 1 strength 2 18

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) ● tile type = unit square ● each side has a glue , with a label and a strength (0,1,2) ● if tiles with matching glues touch, they're strength 0 attracted with the glue's strength strength 1 ● tiles cannot rotate strength 2 19

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) ● tile type = unit square ● finitely many tile types ● each side has a glue , with a label and a strength (0,1,2) ● if tiles with matching glues touch, they're strength 0 attracted with the glue's strength strength 1 ● tiles cannot rotate strength 2 20

Abstract Tile Assembly Model Abstract Tile Assembly Model (Winfree 1998) (Winfree 1998) ● tile type = unit square ● finitely many tile types ● each side has a glue , ● infinitely many tiles : with a label and a copies of each tile type strength (0,1,2) (6.02•10 23 ≈ ∞ ) ● if tiles with matching glues touch, they're strength 0 attracted with the glue's strength strength 1 ● tiles cannot rotate strength 2 21

Self-assembly Main results Conclusions ATAM Models: Producible assemblies Seeded (standard) model: base case: the seed tile recursive case: result of attaching a single tile to a producible assembly if attracted with the total strength of glues at least temperature τ SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions ATAM Models: Producible assemblies Seeded (standard) model: base case: the seed tile recursive case: result of attaching a single tile to a producible assembly if attracted with the total strength of glues at least temperature τ Hierarchical model: base case: any single title recursive case: result of attaching two producible assemblies if attracted with sufficient strength of glues and no space conflict SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2 Overlap disallowed in attachment events (“steric protection”)

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2 Overlap disallowed in attachment events (“steric protection”)

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2 Overlap disallowed in attachment events (“steric protection”)

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2 Overlap disallowed in attachment events (“steric protection”)

Hierarchical tile assembly model Hierarchical Tile Assembly Model Example at Temperature 2 Overlap disallowed in attachment events (“steric protection”)

Self-assembly Main results Conclusions Assembly (definitions) An assembly is a partial map from Z 2 to the set of tiles T . SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Assembly (definitions) An assembly is a partial map from Z 2 to the set of tiles T . dom α is a domain of assembly α , i.e., a subset of positions in the grid that are mapped to tiles. SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Assembly (definitions) An assembly is a partial map from Z 2 to the set of tiles T . dom α is a domain of assembly α , i.e., a subset of positions in the grid that are mapped to tiles. v ∈ Z 2 , α + � Given a vector � v is a translation of α (formally, v )( p ) = α ( p − � ( α + � v )). Note that dom( α + � v ) = dom α + � v . SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Main result about Hierarchical Model Definition We say that assembly α repetitious (has a “pattern overlap”) if v ∈ Z 2 such that there exists a nonzero vector � dom α ∩ dom( α + � v ) � = ∅ and α and α + � v are consistent. SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Main result about Hierarchical Model Definition We say that assembly α repetitious (has a “pattern overlap”) if v ∈ Z 2 such that there exists a nonzero vector � dom α ∩ dom( α + � v ) � = ∅ and α and α + � v are consistent. Theorem Let T be a hierarchical tile assembly system. If T has a producible repetitious assembly, then arbitrarily large assemblies are producible in T . SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Main result about Hierarchical Model Definition We say that assembly α repetitious (has a “pattern overlap”) if v ∈ Z 2 such that there exists a nonzero vector � dom α ∩ dom( α + � v ) � = ∅ and α and α + � v are consistent. Theorem Let T be a hierarchical tile assembly system. If T has a producible repetitious assembly, then arbitrarily large assemblies are producible in T . α 0 (a) SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Main result about Hierarchical Model Definition We say that assembly α repetitious (has a “pattern overlap”) if v ∈ Z 2 such that there exists a nonzero vector � dom α ∩ dom( α + � v ) � = ∅ and α and α + � v are consistent. Theorem Let T be a hierarchical tile assembly system. If T has a producible repetitious assembly, then arbitrarily large assemblies are producible in T . α 0 α 0 α 1 (a) (b) SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems

Self-assembly Main results Conclusions Main result about Hierarchical Model Definition We say that assembly α repetitious (has a “pattern overlap”) if v ∈ Z 2 such that there exists a nonzero vector � dom α ∩ dom( α + � v ) � = ∅ and α and α + � v are consistent. Theorem Let T be a hierarchical tile assembly system. If T has a producible repetitious assembly, then arbitrarily large assemblies are producible in T . β 2 α 0 α 0 α 1 α 2 α 0 α 1 (a) (b) (c) (d) SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems