Ashish Goel, [email protected]1 The Source of Errors: Thermodynamics of correct growth ¼ exp(-G A ) ability of incorrect growth ¼ exp(-G A + G B ) traint: 2 G B > G A (system goes forward) ) Error probability ¸ exp(-G A /2) ) Rate has quadratic dependence on error probability ) Time to reliably assemble an n £ n square ¼ n 5 G A = Activation energy G B = Bond energy G A G B G A 2G B + Correct Growth Incorrect Growth
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Ashish Goel, 1 The Source of Errors: Thermodynamics Rate of correct growth ¼ exp(-G A ) Probability of incorrect growth ¼ exp(-G A.
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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]
•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.
•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
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]
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
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”