the source of errors: thermodynamics
DESCRIPTION
The Source of Errors: Thermodynamics. G A = Activation energy G B = Bond energy. +. 2G B. G B. G A. G A. Correct Growth. Incorrect Growth. Rate of correct growth ¼ exp(-G A ) Probability of incorrect growth ¼ exp(-G A + G B ) Constraint: 2 G B > G A (system goes forward) - PowerPoint PPT PresentationTRANSCRIPT
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
000
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
000
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 – 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 thatq Given a row of 2 log N tiles encoding W(k), there is some simple “tiling
subroutine” to assemble W(k+1) on top
q 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