Nanocomputations by DNA

Self-Assembly

Lila KariDept. of Computer Science, Dept. of Mathematics, Dept. of Biochemistry

University of Western OntarioLondon, ON, Canadahttp://www.csd.uwo.ca/~lila/

lila@csd.uwo.caIn collaboration with L.Adleman, J.Kari, D.Reishus,

P.Sosik

Lila Kari, University of Western Ontario

Self-Assembly• Self-Assembly = The process by which objects

autonomously come together to form complex structures

Lila Kari, University of Western Ontario

Self-Assembly• Self-Assembly = The process by which objects

autonomously come together to form complex structures

• Examples§ Atoms bind by chemical bonds

to form molecules

§ Molecules may form crystals or macromolecules

§ Cells interact to form organisms

Lila Kari, University of Western Ontario

Motivation for Self-Assembly

Nanotechnology: miniaturization in medicine, electronics, engineering, material science, manufacturing

• Top-Down techniques (e.g. litography) are inefficient in creating structures with the size of molecules or atoms

• Bottom-Up techniques: self-assembly

Lila Kari, University of Western Ontario

Applications of Self-Assembly

• Circuit Fabrication• Nanorobotics• DNA Computation• Amorphous Computing

Outline

Lila Kari, University of Western Ontario

(I) A mathematical model for self-assembly

(II) An unsolvable self-assembly problem

(III) How to compute using self-assembly of DNA nanotiles

Lila Kari, University of Western Ontario

Model of Self-Assembly:Tile System [Wang61]

• Tile = square with edges labelled from a finite alphabet of glues

• Tiles cannot be rotated• Two adjacent tiles on the plane match

(stick) if they have the same glue at the touching edges

Lila Kari, University of Western Ontario

Tile System• Tile System T = Finite set of tiles,

unlimited supply of each “tile type”

A B DC

• A tiling (assignment of tiles to points on the integer grid) is valid if adjacent edges of neighbouring tiles have the same glue.A AAA

AA

A

B

BB

B

B

B

B

C

C

C

D

D

C

C

C B

B

Lila Kari, University of Western Ontario

Yes

Classical Tiling Problem

• Can any square, of any size, be tiled using only the available tile types, without violating the glue-matching rule?

NoHarel, D. Computers Ltd. 2000

Lila Kari, University of Western Ontario

Classical “Tiling Problem”“Given a tile system T, does there exist a valid tiling of the plane with tiles from T?’’

The Tiling Problem is undecidable (theredoes not exist an algorithm for solving it)[Berger66], [Robinson71]

Lila Kari, University of Western Ontario

Turing Machine (TM)Model of computation/algorithm/program

* Tape (cells)* Read/write head* States qi ; Input symbols sj

* Rewriting rules qi sj à sk L qn

Halting Problem• Turing Machine – mathematical model

of “program” (algorithm)• Halting Problem: Does there exist a

program (TM) with:Input: A program P and an input IOutput: “yes” if the program P halts on

input I and “no” otherwise• Answer: No. The Halting Problem is

undecidable (unsolvable)Lila Kari, University of Western Ontario

Proof (by contradiction)

• Assume such a TM exists, call it H(P, I) where P is program and I is input

• H outputs “halt” (Y) or “loop forever”(N)• Construct a new program K(P) such that

* If H(P,P) outputs “loop forever” it halts * If H(P, P) outputs “halt” it goes into an infinite loop printing “ha” at each iteration

Lila Kari, University of Western Ontario

The contradiction

Lila Kari, University of Western Ontario

Call K(K)

If K halts on K then H(K,K) outputs“halt” which means K loops forever on K

If K loops forever on input K, then H(K,K) loops foreverwhich means K halts on K.

Lila Kari, University of Western Ontario

Turing Machines and Tilings

• The Tiling Problem is undecidable• Proof - Simulate a TM with tiles• For each Turing Machine rule

qi sj à sk L qn or qi sj à sk R qn

construct tiles that have those rules encoded in the glues on their edges

Lila Kari, University of Western Ontario

Alphabet, Action (qi sj à sk R qn),Merging, General Starting Tiles

sk

qn

s0 s0q0 s0s0s0

sk

sk qi sj

qnsj

qn

sj

qn sj

qn

sj

sk

qn

qi sj

Lila Kari, University of Western Ontario

q0 0 00 1

Simulation of TM Computations by Valid Tilings

Lila Kari, University of Western Ontario

q0 0 00 1

1 0q1 0X

q1

Simulation of TM Computations by Valid Tilings

q00 -> X R q1

Lila Kari, University of Western Ontario

q0 0 00 1

1 0q1 0X

q1

q1 1X 00

q1

Simulation of TM Computations by Valid Tilings

q00 -> X R q1

q10 -> 0 R q1

Lila Kari, University of Western Ontario

q0 0 00 1

1 0q1 0X

q1

q1 1X 00

q1

q2 0X 0Y

q2

Simulation of TM Computations by Valid Tilings

q00 -> X R q1

q10 -> 0 R q1

q11 -> Y L q2

Lila Kari, University of Western Ontario

TM and the Tiling Problem

• The tile system admits a valid tiling of the plane if and only if the computation of Turing Machine never halts when started on a blank tape

• Since the Halting Problem for Turing Machines is undecidable, the Tiling Problem is also undecidable

Lila Kari, University of Western Ontario

Modern Self-Assembly Problems

[Winfree98], [Seeman92]

• What is the minimal number of tile types that can self-assemble into a given shape and nothing else?

• What is the optimal initial concentration of tile types that ensures fastest self-assembly?

• What happens if “bonds” have different strengths?

Lila Kari, University of Western Ontario

Self-Assembly as a Process

• Supertiles self-assemble with tiles from T§ Start with an arbitrary single tile: “seed”§ Proceed by incremental additions of single

tiles that stick

A B D

C A A

C

B

Outline

Lila Kari, University of Western Ontario

(I) A mathematical model for self-assembly

(II) An unsolvable self-assembly problem

(III) How to compute using self-assembly of DNA nanotiles

Lila Kari, University of Western Ontario

A Self-Assembly Problem“Given a tile system T, can arbitrarily large

supertiles self-assemble with tiles from T?”Equivalent to:“Given a tile system T, does there exist an

infinite ribbon of tiles from T?”

x

Lila Kari, University of Western Ontario

Generating Ribbons

Lila Kari, University of Western Ontario

Ribbon, Zipper• Ribbon: Consecutive tiles stick

• Zipper: Any adjacent tiles stick.

Lila Kari, University of Western Ontario

Directed Tiles• Directed Tiles: Tiles with direction

• Directed Paths:

The path may enter a loop or the path may be infinite

Lila Kari, University of Western Ontario

Directed Ribbons and Zippers• Directed Ribbons: The glues and

directions of consecutive tiles must matchZIPPER ERROR

• Directed Zippers:

Undecidability Result • It is undecidable, given an arbitrary tile

system T, whether or not an infinite ribbon can be self-assembled with its tiles. [Adleman, Kari, Kari, Reishus, Sosik,SIAM J. of Computing, 2009]

Proof idea: Reduce the (undecidable) “Tiling Problem” to our problem.

• Step 1: Construction• Step 2: Reduction • Step 3: Simulation

Lila Kari, University of Western Ontario

Lila Kari, University of Western Ontario

A directed tile system (T,d) has thestrong plane-filling property iff• There exists an infinite directed zipper• Any infinite directed zipper covers arbitrarily

large squares

Theorem 1: There exists a directed tile system (A, da) with the strong plane-filling property.

Step 1: Construction

Lila Kari, University of Western Ontario

A Directed Tile System (A,da)with the Strong Plane-filling

Property• Start with directed tiles [JKari,91], resembling

Robinson tiles, that admit only aperiodic tilings of the plane

• The tiles have directions that force any directed zipper to form a fractal-like curvesimilar to Hilbert and Peano curves

Lila Kari, University of Western Ontario

A Plane-filling Directed Zipper

Lila Kari, University of Western Ontario

Step 2: ReductionTheorem 2: Given a directed tile system (T,d), it

is undecidable whether or not there exists aninfinite directed zipper formed with those tiles.

Proof: Reduce the (undecidable) Tiling Problem to our problem. Consider a tile system T.

Construct “sandwich tiles”*Top: tiles from (A, da)*Bottom: tiles from T

Claim: T admits a valid tiling of the plane iff the new set of directed sandwich tiles admits an infinite directed zipper

AT

Lila Kari, University of Western Ontario

Step 3: SimulationTheorem 3: Given a tile system T, it is

undecidable whether or not there exists an infinite ribbon of tiles from T.

Proof: Reduce the (undecidable) infinite-directed-zipper problem of Th2 to the ribbon-problem.

Let (T,d) be a set of directed tiles. Construct a set Tu of undirected tiles such that

• (T,d) admits an infinite-directed-zipper iff• Tu admits an infinite-ribbon

Lila Kari, University of Western Ontario

Construct Undirected Tiles Simulate directed-zipper-tiles byribbon-tile-motifs

zipper-tile d motif of ribbon-tiles

Lila Kari, University of Western Ontario

Ribbon-Motifs

Left entry Bottom entry Right entry

3 ribbon-motifs for a zipper-tile with direction North

Lila Kari, University of Western Ontario

Ribbon-Motifs• Input (output) ribbon-tiles have glues

that match corresponding tiles in other motifs

Lila Kari, University of Western Ontario

Ribbon-Motifs• Paint unpainted sides with non-stick

glues

Edges that do not match any tile

Free ends

Lila Kari, University of Western Ontario

• To each glue corresponds a unique position

Simulating Glues by Geometry

Dent

Bump

• Glue matching is simulated by pairs ofbump-fits-the-corresponding-dent

Lila Kari, University of Western Ontario

By gluing infinitely many ribbon-motifs by their free ends we obtain infinite ribbons that exactly correspond to infinite directed zippers

Lila Kari, University of Western Ontario

By gluing infinitely many ribbon-motifs by their free ends we obtain infinite ribbons that exactly correspond toinfinite directed zippers

Lila Kari, University of Western Ontario

By gluing infinitely many ribbon-motifs from their free ends we obtain infinite ribbons that exactly correspond to infinite directed zippers

Lila Kari, University of Western Ontario

Putting it Together(Th1) Construction: There exists a directed Tile

System with the strong-plane-filling property.

(Th2) Reduction: Given a Tile System, it is undecidable whether or not it can assemble an infinite directed zipper. (Sandwich construction)

(Th3) Simulation: Given a Tile System, it is undecidable whether or not it can assemble an infinite ribbon. ( Ribbon-Motif Construction)

Lila Kari, University of Western Ontario

Sometimes We Cannot Do It!

Theuncomputable(undecidable)

Thecomputable(decidable)

Harel, D. Computers Ltd. 2000

Outline

Lila Kari, University of Western Ontario

(I) A mathematical model for self-assembly

(II) An unsolvable self-assembly problem

(III) How to compute using self-assembly of DNA nanotiles

Lila Kari, University of Western Ontario

How to Make DNA Tiles

DNA

How to Make DNA Tiles

Lila Kari, University of Western Ontario[Chen, Seeman, Winfree, He]

Lila Kari, University of Western Ontario

Lila Kari, University of Western Ontario

Logical Computation byDNA Self-Assembly

(Mao, LaBean, Reif, , Seeman, Nature, 2000)Cumulative XOR: Yi= Yi-1 XOR Xi

Lila Kari, University of Western Ontario

DNA Nanotechnology(Chen, Seeman, Nature, 2001)

Lila Kari, University of Western Ontario

DNA Clonable Octahedron(Shih, Joyce, Nature 2004)

Lila Kari, University of Western Ontario

Nanoscale DNA Tetrahedra(Goodman, Turberfield, Science, 2005)

Triangular DNA tiles

Lila Kari, University of Western Ontario

[LiuWangDengWaluluMao2004], [KariSekiXu2012]

Hexagonal DNA tiles

Lila Kari, University of Western Ontario

[WilnerEtAl2011], [KariSekiXu2012]

[ZhaoLiuYan2011],

3D DNA-Tile Self-Assembly

Lila Kari, University of Western Ontario[KeOngShihYin2012]

Lila Kari, University of Western Ontario

Conclusion(I) Mathematical model for DNA self-assembly –

Wang Tiling Systems(II) Some problems are unsolvable -

Undecidability of the “Infinite Ribbon Problem”[Adleman, Kari, Kari, Reishus, Sosik,

SIAM J.Computing, 2009](III) Wet-lab experiments with DNA nanotiles

- DNA computations (variable glue strengths)- Different DNA tile shapes – triangle, hexagon- Three-dimensional DNA self-assembly