the source of errors: thermodynamics

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 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, ashishg@stanford.edu 3

Example: Sierpinski Tile System

00

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 4

Example: Sierpinski Tile System

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

000

00

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 6

Example: Sierpinski Tile System

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 7

Growth Error

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

01

10

Ashish Goel, ashishg@stanford.edu 8

Growth Error

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

mismatch

01

10

Ashish Goel, ashishg@stanford.edu 9

Growth Error

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

010

Ashish Goel, ashishg@stanford.edu 10

Growth Error

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 11

Growth Error

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 13

How does this help?

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 14

How does this help?

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

mismatch

Ashish Goel, ashishg@stanford.edu 15

How does this help?

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 17

How does this help?

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 18

How does this help?

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 19

How does this help?

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

Ashish Goel, ashishg@stanford.edu 20

Nucleation Error

Ashish Goel, ashishg@stanford.edu 21

Nucleation Error

•First tile attaches with a weak binding strength

Ashish Goel, ashishg@stanford.edu 22

Nucleation Error

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

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 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, ashishg@stanford.edu 25

How does this help?

•First tile attaches with a weak binding strength

Ashish Goel, ashishg@stanford.edu 26

How does this help?

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

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 28

Preliminary Experimental Results

(Obtained by Chen, Goel, Schulman, Winfree)

Ashish Goel, ashishg@stanford.edu 29

Ashish Goel, ashishg@stanford.edu 30

Ashish Goel, ashishg@stanford.edu 31

Ashish Goel, ashishg@stanford.edu 32

Ashish Goel, ashishg@stanford.edu 33

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 34

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 35

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 36

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 37

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 38

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 39

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 40

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 41

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 42

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 43

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 44

Four by Four Snake Tiles

Ashish Goel, ashishg@stanford.edu 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, ashishg@stanford.edu 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, ashishg@stanford.edu 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, ashishg@stanford.edu 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, ashishg@stanford.edu 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, ashishg@stanford.edu 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