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 Ma Arash Rafiey 4 , 5 Ladislav Stacho 4 1 National Taiwan University, Taiwan 2
Self-assembly Main results Conclusions
Ho-Lin Chen1 David Doty2 J´ an Maˇ nuch3,4 Arash Rafiey4,5 Ladislav Stacho4
1National Taiwan University, Taiwan 2California Institute of Technology, USA 3University of British Columbia, Canada 4Simon Fraser University, Canada 5Indiana 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
Engineering goal
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
7
DNA
8
Journal of Theoretical Biology 1982)
sticky end
DNA tile
9
Journal of Theoretical Biology 1982)
sticky end
DNA tile
10
11
Liu, Zhong, Wang, Seeman, Angewandte Chemie 2011
12
Using more types of tiles (i.e., different sticky ends)...
Wei, Dai, Yin, Nature 2012
15
(Winfree 1998)
16
(Winfree 1998)
17
strength 1 strength 0 strength 2
with a label and a strength (0,1,2)
(Winfree 1998)
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strength 1 strength 0 strength 2
with a label and a strength (0,1,2)
glues touch, they're attracted with the glue's strength
(Winfree 1998)
19
strength 1 strength 0 strength 2
with a label and a strength (0,1,2)
glues touch, they're attracted with the glue's strength
(Winfree 1998)
20
strength 1 strength 0 strength 2
with a label and a strength (0,1,2)
glues touch, they're attracted with the glue's strength
(Winfree 1998)
21
strength 1 strength 0 strength 2
with a label and a strength (0,1,2)
glues touch, they're attracted with the glue's strength
copies of each tile type (6.02•1023 ≈ ∞)
(Winfree 1998)
Self-assembly Main results Conclusions
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
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
Example at Temperature 2
Example at Temperature 2
Example at Temperature 2
Example at Temperature 2
Example at Temperature 2
Overlap disallowed in attachment events (“steric protection”)
Example at Temperature 2
Overlap disallowed in attachment events (“steric protection”)
Example at Temperature 2
Overlap disallowed in attachment events (“steric protection”)
Example at Temperature 2
Overlap disallowed in attachment events (“steric protection”)
Example at Temperature 2
Overlap disallowed in attachment events (“steric protection”)
Example at Temperature 2
Self-assembly Main results Conclusions
An assembly is a partial map from Z2 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
An assembly is a partial map from Z2 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
An assembly is a partial map from Z2 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. Given a vector v ∈ Z2, α + v is a translation of α (formally, (α + v)(p) = α(p − 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
Definition We say that assembly α repetitious (has a “pattern overlap”) if there exists a nonzero vector v ∈ Z2 such that 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
Definition We say that assembly α repetitious (has a “pattern overlap”) if there exists a nonzero vector v ∈ Z2 such that 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
Definition We say that assembly α repetitious (has a “pattern overlap”) if there exists a nonzero vector v ∈ Z2 such that 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 .
α0 (a)
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Definition We say that assembly α repetitious (has a “pattern overlap”) if there exists a nonzero vector v ∈ Z2 such that 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 .
α0 (a) α0α1 (b)
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Definition We say that assembly α repetitious (has a “pattern overlap”) if there exists a nonzero vector v ∈ Z2 such that 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 .
α0 (a) α0 α1 α2 (c) α0α1 (b) (d) β2
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Main Tool
Theorem (Doty (UCNC 2014) — Main Tool) If two producible assemblies α and β overlap and are consistent, then α ∪ β is producible. Moreover, α ∪ β can be produced that α is assembled first followed by the missing portions of β \ α.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Main Tool
Theorem (Doty (UCNC 2014) — Main Tool) If two producible assemblies α and β overlap and are consistent, then α ∪ β is producible. Moreover, α ∪ β can be produced that α is assembled first followed by the missing portions of β \ α.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Main Tool
Theorem (Doty (UCNC 2014) — Main Tool) If two producible assemblies α and β overlap and are consistent, then α ∪ β is producible. Moreover, α ∪ β can be produced that α is assembled first followed by the missing portions of β \ α.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Let αi = α + i v Assumptions: α0 and α1 overlap and are consistent
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Let αi = α + i v Assumptions: α0 and α1 overlap and are consistent Easy case: If αi for all integers i ≥ 2 does not overlap α0, we can keep adding copies of α, and the Main Tool will guarantee producibility:
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Let αi = α + i v Assumptions: α0 and α1 overlap and are consistent Easy case: If αi for all integers i ≥ 2 does not overlap α0, we can keep adding copies of α, and the Main Tool will guarantee producibility: α0
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Let αi = α + i v Assumptions: α0 and α1 overlap and are consistent Easy case: If αi for all integers i ≥ 2 does not overlap α0, we can keep adding copies of α, and the Main Tool will guarantee producibility: α0 ∪ α1
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Let αi = α + i v Assumptions: α0 and α1 overlap and are consistent Easy case: If αi for all integers i ≥ 2 does not overlap α0, we can keep adding copies of α, and the Main Tool will guarantee producibility: α0 ∪ α1 ∪ α2
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Let αi = α + i v Assumptions: α0 and α1 overlap and are consistent Easy case: If αi for all integers i ≥ 2 does not overlap α0, we can keep adding copies of α, and the Main Tool will guarantee producibility: α0 ∪ α1 ∪ α2 ∪ α3
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
α0 (a) α0α1 (b)
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
α0 (a) α0 α1 α2 (c) α0α1 (b)
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
α0 (a) α0 α1 α2 (c) α0α1 (b) (d) β2
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0.
domα2 domα1 domα0
⃗
v C2 C3 C1 C4
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious:
α′ is producible by the Main Tool Theorem
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious:
α′ is producible by the Main Tool Theorem We have that dom α′ ∩ dom(α′ + v) = (dom α0 ∩ dom α1) ∪ (C − v)
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious:
α′ is producible by the Main Tool Theorem We have that dom α′ ∩ dom(α′ + v) = (dom α0 ∩ dom α1) ∪ (C − v) disjoint
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious:
α′ is producible by the Main Tool Theorem We have that dom α′ ∩ dom(α′ + v) = (dom α0 ∩ dom α1) ∪ (C − v) consistent since α0 = α and α1 = α + v are consistent
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious:
α′ is producible by the Main Tool Theorem We have that dom α′ ∩ dom(α′ + v) = (dom α0 ∩ dom α1) ∪ (C − v) consistent by definition of α′
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0. Let C be a component of dom α2 \ dom α1 that does not
Then C − v is a component of dom α1 \ dom α0 Let α′ = α0 ∪ α1↾C−
v (restriction). We will show that α′ is
producible and repetitious | dom α′| > | dom α|. We can repeat this process to find arbitrary large producible assemblies.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof
Lemma dom α2 \ dom α1 has a (maximal connected) component that does not overlap dom α0.
Domα' +v Domα' ⃗ v
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Assume to the contrary that every component Ci of dom α2 \ dom α1 intersects dom α0 at point pi.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Assume to the contrary that every component Ci of dom α2 \ dom α1 intersects dom α0 at point pi. Let ni be the smallest integer such that pi + ni v / ∈ dom α1. Then pi and pi + ni v lie in dom α2 \ dom α1.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Assume to the contrary that every component Ci of dom α2 \ dom α1 intersects dom α0 at point pi. Let ni be the smallest integer such that pi + ni v / ∈ dom α1. Then pi and pi + ni v lie in dom α2 \ dom α1. If they would lie in the same component Cj, then there is a path p between them in dom α2 \ dom α1. The translation p − v lies in dom α1, so it can’t intersect p. This contradicts the following lemma: Lemma (Cannon et al. (STACS 2013)) Let φ be a curve connecting point p and p + n v in R2, where n is an
v.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Assume to the contrary that every component Ci of dom α2 \ dom α1 intersects dom α0 at point pi. Let ni be the smallest integer such that pi + ni v / ∈ dom α1. Then pi and pi + ni v lie in dom α2 \ dom α1. Hence, each pi and pi + ni v lie in different components of dom α2 \ dom α1.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Assume to the contrary that every component Ci of dom α2 \ dom α1 intersects dom α0 at point pi. Let ni be the smallest integer such that pi + ni v / ∈ dom α1. Then pi and pi + ni v lie in dom α2 \ dom α1. Hence, each pi and pi + ni v lie in different components of dom α2 \ dom α1. Since the number of components is finite, there must exists a sequence pj1, . . . , pjk such that pji and pji+1 + nji+1 v lie in the same (unique) component, where indices are taken modulo k.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Since the number of components is finite, there must exists a sequence p1, . . . , pk such that pi and pi+1 + ni+1 v lie in the same (unique) component, where indices are taken modulo k.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Since the number of components is finite, there must exists a sequence p1, . . . , pk such that pi and pi+1 + ni+1 v lie in the same (unique) component, where indices are taken modulo k. There are curves ϕi connecting pi and pi+1 + ni+1 v lying inside its component.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Since the number of components is finite, there must exists a sequence p1, . . . , pk such that pi and pi+1 + ni+1 v lie in the same (unique) component, where indices are taken modulo k. There are curves ϕi connecting pi and pi+1 + ni+1 v lying inside its component. Since these curves lie in different components, they do not intersect.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Lemma dom α2 \ dom α1 has a component that does not overlap dom α0.
Proof. Since the number of components is finite, there must exists a sequence p1, . . . , pk such that pi and pi+1 + ni+1 v lie in the same (unique) component, where indices are taken modulo k. There are curves ϕi connecting pi and pi+1 + ni+1 v lying inside its component. Since these curves lie in different components, they do not intersect. Moreover, ϕi and ϕi + v do not intersect, since ϕi + v lies in dom α3 \ dom α2.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Conditions: There are curves ϕi connecting pi and pi+1 + ni+1 v lying inside its component, where ni+1 ∈ N. The curves do not intersect. ϕi and ϕi + v do not intersect. Example:
p1 p2+ v p2 p3+ v p3 p4+ v p4 p1+ 3
5
v ϕ1 ϕ2 ϕ3 ϕ4
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Proof of Lemma
Conditions: There are curves ϕi connecting pi and pi+1 + ni+1 v lying inside its component, where ni+1 ∈ N. The curves do not intersect. ϕi and ϕi + v do not intersect. Example:
p1 p2+ v p2 p3+ v p3 p4+ v p4 p1+ 3
5
v ϕ1 ϕ2 ϕ3 ϕ4
But n1 = 3/5 / ∈ N.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
Theorem For any set of points in R2, p1, . . . , pk, non-zero vector v ∈ R2 and integers n1, . . . , nk ∈ N, there do not exist curves ϕ1, . . . , ϕk satisfying the following conditions:
1 ϕi connects pi and pi+1 + ni+1
v.
2 The curves do not intersect. 3 ϕi and ϕi +
v do not intersect.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
We have proved that if a producible assembly of a tile system “consistently overlaps” itself, then it can produce arbitrary large assemblies and therefore cannot produce a unique finite assembly.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
We have proved that if a producible assembly of a tile system “consistently overlaps” itself, then it can produce arbitrary large assemblies and therefore cannot produce a unique finite assembly. Observations:
1 This result does not apply to the Seeded Model.
Example: (Source: wikipedia)
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems
Self-assembly Main results Conclusions
We have proved that if a producible assembly of a tile system “consistently overlaps” itself, then it can produce arbitrary large assemblies and therefore cannot produce a unique finite assembly. Observations:
1 This result does not apply to the Seeded Model. 2 [Chen, Doty (SODA 2012)] has showed that if a tile system
does not produce assemblies that consistently overlap any translation of themselves, then it cannot produce any shape in time sublinear in its diameter. Together with our result: any tile system that produces a unique finite assembly in a (hierarchical) tile system requires time Ω(d), where d is the diameter of the finite assembly. Hence, despite the parallelism of the Hierarchical Model, it cannot produce assemblies faster than the Seeded Model.
SoCG 2015 J´ an Maˇ nuch Pattern overlap implies runaway growth in hierarchical tile systems