ashish goel, 1 the source of errors: thermodynamics rate of correct growth ¼ exp(-g a ) probability...
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Ashish Goel, 3 Example: Sierpinski Tile SystemTRANSCRIPT
Ashish Goel, [email protected] 1
The Source of Errors: Thermodynamics
Rate of correct growth ¼ exp(-GA)
Probability of incorrect growth ¼ exp(-GA + GB)
Constraint: 2 GB > 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
GA = Activation energy
GB = Bond energy
GA
GBGA
2GB
+
Correct Growth Incorrect Growth
Ashish Goel, [email protected] 2
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]
Ashish Goel, [email protected] 4
Example: Sierpinski Tile System
0
1
1
1
0
0
0
0
1
0
1
1
1
1
0
0 00
00
Ashish Goel, [email protected] 5
Example: Sierpinski Tile System
0
1
1
1
0
0
0
0
1
0
1
1
1
1
0
0 00
00 0
Ashish Goel, [email protected] 12
Proofreading Tiles
• Each tile in the original system corresponds to four tiles in the new system
• The internal glues are unique to this block
G1
G4
G3
G2
G1b
X4
X3
G2a
X2
G3b
G2b
G1a
G4a
X1
G4b
G3a
[Winfree, Bekbolatov, ’03]
Ashish Goel, [email protected] 16
How does this help?
0
1
1
1
0
0
0
0
1
0
1
1
1
1
0
0
No tile can attachat this location
Ashish Goel, [email protected] 21
Nucleation Error
•First tile attaches with a weak binding strength
Ashish Goel, [email protected] 22
Nucleation Error
•First tile attaches with a weak binding strength•Second tile attaches and secures the first tile
Ashish Goel, [email protected] 23
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.
Ashish Goel, [email protected] 24
Snake Tiles
• Each tile in the original system corresponds to four tiles in the new system
• The internal glues are unique to this block
G1
G4
G3
G2
G1b
X1
X2
G2a
X3
G3b
G2b
G1a
G4a G4b
G3a
Ashish Goel, [email protected] 25
How does this help?
•First tile attaches with a weak binding strength
Ashish Goel, [email protected] 26
How does this help?
•First tile attaches with a weak binding strength•Second tile attaches and secures the first tile
Ashish Goel, [email protected] 27
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
Ashish Goel, [email protected] 28
Preliminary Experimental Results
(Obtained by Chen, Goel, Schulman, Winfree)
Ashish Goel, [email protected] 29
Ashish Goel, [email protected] 30
Ashish Goel, [email protected] 31
Ashish Goel, [email protected] 32
Ashish Goel, [email protected] 45
Analysis Snake tile design extends to 2k£2k blocks.
Prevents tile propagation even after k+1 nucleation/growth errors The error probability changes from p to roughly pk
We can assemble an N£N square in time O(N polylog N) and it remains stable for time (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 Reif’s group, Winfree’s group, our
group, and others Recent result: Simple combinatorial criteria; Can avoid resolution loss
by using third dimension [Chen, Goel, Luhrs; SODA ‘08]
Ashish Goel, [email protected] 46
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 > 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
Ashish Goel, [email protected] 47
Interesting Open Problems – IIRobust, 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 thatq Given a row of 2 log N tiles encoding W(k), there is some simple “tiling
subroutine” to assemble W(k+1) on topq 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
Ashish Goel, [email protected] 48
(1,1)(1,0)(0,1)(0,0)(1,1)
(0,1)(1,1)(0,1)(1,1)(0,1)
(0,1)
(1,0)
(0,0)
(1,1)
(1,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0) (0,0)
(0,0)
(0,1) (1,1) (0,0)
(1,1)
(1,0)
(1,1)
Another Mode of Error -- Damage
1W
1W
1W
1W
1W
1S1S1S1S1S1S
(1,1)
(1,0) (0,1)
(1,1)
(0,0)
(1,0)
S
S
1W
1S
(0,0)
(0,1)
(1,1)
(1,0)
Ashish Goel, [email protected] 49
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”
Ashish Goel, [email protected] 50
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