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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
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