DNA computing and self- -assembly assembly DNA computing and self - - PowerPoint PPT Presentation

DNA computing and self DNA computing and self-

• assembly

assembly

Jie Gao

References: Erik Winfree and Ashish Goel

Computing Computing

Human technology

Electronic circuits.

Biological systems

Biochemical circuits.

Motivation for DNA computing

what biochemical algorithms are capable of. How to program biochemical circuits.

State of the art

Theory: design algorithms Experiments: synthetic DNA molecules

A little bit history on tiling A little bit history on tiling

• Tile the infinite plane.
• Periodic tiling.
• Aperiodic tiling.

Question Question

• Given a set of tiles, can we decide

whether there is a way to tile the entire plane?

• Answer: not solvable (Wang 1963).
• Hint: it is equivalent to a Turing machine.

– If the Turing machine halts, then no way to tile. – If the Turing machine computes forever, then a global tiling is possible.

• Then, can we use this for computing?

A counter by tiling A counter by tiling

Using tiling for computing Using tiling for computing

• Two immediate questions:

– Implement Wang tiles. – Rules to have the tiles grow reliably.

• DNA nanotechnology (Seeman 1982)

– Implement the tiles by synthetic DNA molecules.

DNA Tiles DNA Tiles

Glues = sticky ends Tiles = molecules

[Winfree]

Tile System: Tile System:

Temperature: A positive integer. A set of tile types: Each tile is an oriented square with glues on its

• edges. Each glue has a non-negative strength.

Infinite supply of tiles. An initial assembly (seed).

A tile can attach to an assembly iff the combined strength of the “matchings glues” is greater or equal than the temperature τ.

[Rothemund, Winfree, ’2000]

Example of a tile system Example of a tile system

Temperature: 2 Set of tile types: Seed:

Example of a SA process Example of a SA process

Temperature: 2 Set of tile types: Seed:

Example of a SA process Example of a SA process

Temperature: 2 Set of tile types: Seed:

Example of a SA process Example of a SA process

Temperature: 2 Set of tile types: Seed:

Tile Assembly: a counter Tile Assembly: a counter

• Oriented tiles with glue on each side.
• Each side: a label, and a strength.
• To assemble, total strength τ =2.

1 2 No glue

The counter tiles The counter tiles

• Input: bottom and right.
• Output: top and left.

input carry

• utput

Output carry

The counter tiles The counter tiles

• Final tiling is unique
• Parallel computing

Another example: parity check Another example: parity check

• Four parity check tiles.

Why DNA assembly? Why DNA assembly?

• Practical reasons: it is small & just some

molecule.

• Potentially can be injected into human body to

cure diseases.

• Fabrication: automatically generate certain

nano-scale patterns (say, a CPU – a long-term goal).

• Solve NP-hard problems (due to inherent

parallelism).

Analysis of DNA self Analysis of DNA self-

• assembly

assembly

• The program size:

– the number of tiles with different type. – Infinite supply of each tile type.

• Time complexity:

– How long it takes to finish the self-assembly. – We will talk about this later.

• How does the strength for binding matter here?
• First question: grow something manageable.

– given a desired shape, how to design tiles to implement it?

A few easy things A few easy things

• Assemble an N-bit long line

– Program size =Θ(N), why? – Strength for binding, τ>0. – Not very efficient.

• Counter tiles.

– Program size = Θ(1) – Strength for binding, τ =2.

• Harder: Assemble an N by N square.

Assemble an N by N square Assemble an N by N square

• τ=1, Naïve program.

– Number of tiles = N2.

Assemble an N by N square Assemble an N by N square

• τ=1, Naïve program is the optimal.

– Number of tiles = N2.

Assemble an N by N square Assemble an N by N square

• τ=2, things are remarkably different.

– Think about the counter. – How to stop it from growing to infinity? – We know how to assemble a 1D line. – Program size = O(N). – Can you do better?

Assemble an N by N square Assemble an N by N square

• τ=2, program size=O(N). Slightly better.

A counter in unary. Replace by a counter in binary.

Assemble an N by N square Assemble an N by N square

• τ=2,
• program

size=O(logN).

Assemble an N by N square Assemble an N by N square

• Can you do even better?
• Program size =

O(log*N)

N by N

logN by logN loglogN by loglogN

This, however, depends

• n what N is.

Assemble an N by N square Assemble an N by N square

• You can still do better! --- for some N.
• Good news: There is a sequence of N,

s.t. the program size is O(f(N)), for any increasing function f(N) that goes to infinity.

• Bad news: There is a sequence of N,

s.t. the program size is Ω(logN/loglogN).

Summary of computability Summary of computability

• 1D self-assembly: not much you can do.
• Strength τ=2, 2D self-assembly is universal.
• 3D self-assembly: likely nothing interesting

either. A DNA “rug”, 500 nm

wide, and is assembled using DNA tiles roughly 12nm by 4nm. (Erik Winfree, Caltech)

Tile Systems and Running Time Tile Systems and Running Time Seed

A B C

τ = 2

Tile Systems and Running Time Tile Systems and Running Time

Seed Seed

A B C

Seed

A

Seed

B

Seed

A B

A: 50%, B:30%, C: 20% 0.5 0.3 0.2 0.3 0.5

Tile Systems and Running Time Tile Systems and Running Time

• Define a continuous time Markov chain

– State space : set of all structures that a tile system can assemble into – Tiles of type have concentration

• Σ = 1

– Unique terminal structure => unique sink in – Seed tile is the unique source state

• Assembly time: the hitting time for the sink state

from the source state in the Markov Chain

Old Example: Assembling Lines Old Example: Assembling Lines

• Very simple Markov Chain. Average time

for the th tile to attach is 1/

– Assembly time = 1/

• For fastest assembly, all tiles must have

the same concentration of 1/(-1)

– Expected assembly time is – Can assemble thicker rectangles much faster and with much fewer different tiles

Assemble a square, running time Assemble a square, running time

• O(logN) tiles,

each tile has density 1/logN.

• Intuition: grow

some linear lines.

• Fill out the

blanket in parallel.

Robustness Robustness

• In practice, self-assembly is a

thermodynamic process. When T=2, tiles with 0 or 1 matches also attach; tiles held by total strength 2 also fall off at a small rate.

• Currently, there are 1-10% errors observed in

experimental self-assembly.

[Winfree, Bekbolatov, ’03]

• Possible schemes for error correction

– Biochemistry tricks – Coding theory and error correction

Strand Invasion (cont)

Example Example

T=2

Example Example

T=2

Example Example

T=2

What can go wrong? What can go wrong?

T=2

What can go wrong? What can go wrong?

T=2

Why it may not matter: Why it may not matter:

T=2

Why it may not matter: Why it may not matter:

T=2

What can go What can go really really wrong? wrong?

T=2

What can go What can go really really wrong? wrong?

T=2

What can go What can go really really wrong wrong? ?

T=2

Safe attachments and invadable systems Safe attachments and invadable systems

Definition: A tile system is an invadable system iff for all assemblies that can be grown from the initial assembly, all possible attachments are safe.

Safe Unsafe

Error Correcting Tile Sets Error Correcting Tile Sets

• Do not change the tile model
• Tile systems are designed to have error

correction mechanisms.

• Two most common errors

– Crystallization error – Nucleation error

Example: Sierpinski Tile System Example: Sierpinski Tile System

1 1 1 1 1 1 1 1

Example: Sierpinski Tile System Example: Sierpinski Tile System

1 1 1 1 1 1 1 1

Example: Sierpinski Tile System Example: Sierpinski Tile System

1 1 1 1 1 1 1 1

Example: Sierpinski Tile System Example: Sierpinski Tile System

1 1 1 1 1 1 1 1

Crystallization Error Crystallization Error

1 1 1 1 1 1 1 1 1 1

Crystallization Error Crystallization Error

1 1 1 1 1 1 1 1

mismatch

1 1

Crystallization Error Crystallization Error

1 1 1 1 1 1 1 1 1

Crystallization Error Crystallization Error

1 1 1 1 1 1 1 1

Crystallization Error Crystallization Error

1 1 1 1 1 1 1 1

Proofreading Tiles 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]

How does this help? How does this help?

1 1 1 1 1 1 1 1

How does this help? How does this help?

1 1 1 1 1 1 1 1

mismatch

How does this help? How does this help?

1 1 1 1 1 1 1 1

How does this help? How does this help?

1 1 1 1 1 1 1 1

No tile can attach at this location

How does this help? How does this help?

1 1 1 1 1 1 1 1

How does this help? How does this help?

1 1 1 1 1 1 1 1

How does this help? How does this help?

1 1 1 1 1 1 1 1

Nucleation Error Nucleation Error

Nucleation Error Nucleation Error

• First tile attaches with a weak binding strength

Nucleation Error Nucleation Error

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

Nucleation Error 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.

Snake Tiles 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

How does this help? How does this help?

• First tile attaches with a weak binding strength

How does this help? How does this help?

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

How does this help? 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

Summary Summary

• Distributed growth of tiles can be used for

computing.

• New definition of how to compute, running

time, program size.

Plans Plans

• Next class: presentations.
• Those who do not have a chance to

present: write a critique for one paper in the list.

• Next week: final project presentation.
• Show your results.

Final report Final report

• Due Dec 22nd.
• 80% grade based on report.
• What to put in the report?

– Your problem, motivation. – Related work. – Your results. – Future work. – Who did what? (for group projects).