The Source of Errors: Thermodynamics

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The Source of Errors Thermodynamics
GA Activation energy GB Bond energy

2GB
GB
GA
GA
Correct Growth
Incorrect Growth
Rate of correct growth ¼ exp(-GA) Probability of
incorrect growth ¼ exp(-GA GB) Constraint 2 GB
gt GA (system goes forward) ) Error probability
exp(-GA/2) ) Rate has quadratic dependence on
error probability ) Time to reliably assemble
an n n square ¼ n5
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Error-Reducing Designs

• Error correction via redundancy do not change
  the model
• Tile systems are designed to have error
  correction mechanisms
• The Electrical Engineering approach -- error
  correcting codes
• But can not use existing coding/decoding
  techniques
• Proofreading tiles Winfree, Bekbolatov,03
• Snake tiles Chen, Goel 04
• Biochemistry techniques
• Strand Invasion mechanism
• Chen, Cheng, Goel, Huang, Moisset de espanes,
  04

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Example Sierpinski Tile System
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Example Sierpinski Tile System
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Example Sierpinski Tile System
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Example Sierpinski Tile System
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Growth Error
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Growth Error
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mismatch
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Growth Error
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Growth Error
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Growth Error
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Proofreading Tiles
Winfree, Bekbolatov, 03
G2
G3
G1
G2a
G2b
G4
X3
G3b
G1b
X4
X2

• Each tile in the original system corresponds to
• four tiles in the new system
• The internal glues are unique to this block

X1
G3a
G1a
G4a
G4b
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How does this help?
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How does this help?
1
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mismatch
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How does this help?
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How does this help?
No tile can attach at this location
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How does this help?
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How does this help?
1
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How does this help?
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Nucleation Error
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Nucleation Error

• First tile attaches with a weak binding strength

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Nucleation Error

• First tile attaches with a weak binding strength
• Second tile attaches and secures the first tile

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Nucleation Error

• First tile attaches with a weak binding strength
• Second tile attaches and secures the first tile
• Other tiles can attach and forms a layer of
  (possibly incorrect) tiles.

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Snake Tiles
G2
G3
G1
G2a
G2b
G4
X2
G3b
G1b
X1
X3

• Each tile in the original system corresponds to
  four tiles in the new system
• The internal glues are unique to this block

G3a
G1a
G4a
G4b
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How does this help?

• First tile attaches with a weak binding strength

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How does this help?

• First tile attaches with a weak binding strength
• Second tile attaches and secures the first tile

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How does this help?

• First tile attaches with a weak binding strength
• Second tile attaches and secures the first tile
• No Other tiles can attach without another
  nucleation error

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Preliminary Experimental Results (Obtained by
Chen, Goel, Schulman, Winfree)
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Four by Four Snake Tiles
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Analysis

• Snake tile design extends to 2k2k blocks.
• Prevents tile propagation even after k1
  nucleation/growth errors
• The error probability changes from p to roughly
  pk
• We can assemble an NN square in time O(N polylog
  N) and it remains stable for time W(N) (with high
  probability).
• Resolution loss of O(log N)
• Assuming tiles held by strength 3 do not fall off
• Matches the time for ideal, irreversible assembly
• Compare to N3 for basic proof-reading and N5 with
  no error-correction in the thermodynamic model
  Chen, Goel DNA 04
• Extensions, variations by Reifs group, Winfrees
  group, our group, and others
• Recent result Simple combinatorial criteria Can
  avoid resolution loss by using third dimension
  Chen, Goel, Luhrs SODA 08

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Interesting Open Problems - I

• General theorems for analyzing reversible
  self-assembly?
• Example Imagine you are given an L, with each
  arm being length N
• From each convex corner, a tile can fall off at
  rate r
• At each concave corner, a tile can attach at
  rate f gt r
• What is the first time that the (N,N) location is
  occupied?
• We believe that the right answer is O(N), can
  prove O(N log N)
• General theorems which relate the combinatorial
  structure of an error-correction scheme to the
  error probability?
• We have combinatorial criteria for error
  correction, but they are not all encompassing

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Interesting Open Problems II

• Robust, efficient counting
• We replace a tile by a k k block, where k ! 1
  as N ! 1
• Or, by a k 1 block if we use the third
  dimension
• Codes (eg. Reed-Solomon) can do much better
• Can we use codes to design more efficient
  counters?
• Specifically Do there exist one-to-one functions
  (code-words)
• W 1,..N ! 1..N2 such that
• Given a row of 2 log N tiles encoding W(k), there
  is some simple tiling subroutine to assemble
  W(k1) on top
• Even if there are p log N errors in the tiling
  process for each row, this process stops after
  counting from 1 to N
• Motivation Correctly assembling large shapes
  up-to molecular precision will be a new
  engineering paradigm so an exciting opportunity
  for theoreticians

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Another Mode of Error -- Damage
(0,1)
S
(1,1)
(1,0)
1W
(0,0)
(1,0)
(0,1)
(0,0)
1S
(1,0)
(1,1)
(1,1)
S
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What went wrong?

• When tiles attach from unexpected directions the
  correct tile is not guaranteed.
• Potential fix Design systems more carefully so
  that the system can reassemble from small pieces
  all over.
• Previous work Winfree 06 Rectilinear Systems
  that will grow back correctly as long as the seed
  remains in place by forcing growth only from the
  seed direction.
• Single point of failure Lose the seed and the
  structure cannot regrow
• Akin to a lizard regenerating a limb
• Our goal Tile systems that heal from small
  fragments anywhere
• Akin to two parts of a starfish growing into
  complete separate starfish
• Almost a reproductive property

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Two pieces of self-healing Immutability and
Progressiveness

• Immutability Only correct tiles may attach.
• (As opposed to the Sierpinski example.)
• Progressiveness Eventually, all tiles attach.
• (Provided one of a set of pieces containing
  enough information remains)
• Example The Chinese remainder counter is almost
  self-healing from any row