Robust Self-Assembly of DNA by MSeYaSwI

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									Robust Self-Assembly
of DNA

              Eduardo Abeliuk
      Dept. of Electrical Engineering
                  Stanford University


                     November 30, 2006
    Agenda for today
   Robust Self-Assembly: definitions and motivation
   Basic assembly model and examples

   “Complexity of Self-Assembled Shapes”
     D. Soloveichik, E.Winfree. DNA Computers 10, LNCS v.3384, 2005
   “Error Free Self-Assembly using Error Prone Tiles”,
     H. Chen, A. Goel. 10th Int. Meeting on DNA Based Computers, 2004.
   “Self-Healing Tile Sets”
     E. Winfree, Nanotechnolgy: Science and computation, p.55-78, 2006
Self-Assembly Theory
   Self-assembly: no precise general definition
   But roughly speaking:
    “ process by which an organized structure can spontaneously
       form from simpler parts”

       Programming
       Complexity
       Fault-tolerance
       Self-healing
       Self-reproduction and evolution

                       Schulman R., Winfree E., “Self-replication and evolution of DNA crystals” 2005.
Self-assembly
   Already present in nature
     Inside cells
     Robust self-assembly of organisms over 18 orders of
       magnitude in volume!
   Bottom-up fabrication of complex structures:
     Arbitrary shapes can be self-assembled (2D)
     Enabled by DNA nanotechnology




               Rothemund PWK, “Folding DNA to create nanoscale shapes and patterns”, Nature 2006
Rothemund PWK, “Folding DNA to create nanoscale shapes and patterns”, Nature 2006
Rothemund PWK, “Folding DNA to create nanoscale shapes and patterns”, Nature 2006
Another Motivation
   Compute “along the way”
     The self-assembly of a crystal can resemble a
      program that leaves the traces of its operations
      embedded in it.
                                                     output:
     input:               output:     input:
01001101011            01001101011




       The assembly of a 2D crystal can simulate a universal
        Turing machine!
Robust self-assembly of DNA
   Do we need robustness?
    "In theory, there is no difference between theory and
       practice. But, in practice, there is."
                             -Jan L.A. van de Snepscheut


   Computing with DNA, and not transistors??
                                  DNA      Current computer
Information density (bits/nm3)     ~1           ~10-11
Parallelism (operations/sec)      ~1018         ~1012
Energy expediture (J/operation)   ~10-19         ~10-9
The tile assembly model
   Infinite lattice:
       ZxZ

Every position in
                                      N
the grid has a relative
position associated:              W       E

  N(i,j)=(i,j+1)                      S

  S(i,j)=(i,j-1)
  E(i,j)=(i+1,j)
  W(i,j)=(i-1,j)
                          (i,j)
Bond types and Tile Types
   Our fundamental unit is a square tile with labelled edges, or
    bond types.        A        B       D
                        B       C   A           B   A           C
                            D               D               D
   We consider a set of bond types . (e.g., ={A,B,C,D,null})
   A reflection or rotation gives a different tile.
    So a tile type is a quadruple:
    and we have unlimited supply of them
   Tiles types with identical edges can pair with each other.
                                    B           A
                                A       B B         C
                                    D           D

   We will represent tile types with different colors. All tile types
    for the set T.        A       B        D
                        B       C       A       B       A       C
                            D               D               D
Tiles
   A tile is a pair               ,
    i.e., it corresponds to a tile with certain tile type
    located in a certain position in our grid
   A configuration is a set of tiles, such that there is
    exactly one tile in every location




         Configuration 1                   Configuration 2
Interaction between tiles
   A strength function
    defines the interactions between two tiles.
   We say a tile t1 interacts with its neighbor t2
    with strength
            g A B C D null
            A 1 0 0 0 0           B         A         A
            B 0 1 0 0 0       A       B B       C C       C
                                  C         D         D
            C 0 0 2 0 0
            D 0 0 0 1 0
           null 0 0 0 0 0

   Usually, only diagonal strength functions are
    considered, and the range of g is {0,1,2}
The tile assembly model (aTAM)
   A tile system is a quadruple               i.e., it consist of
       a set of tile types
       a seed tile
       a strength function
       a binding threshold or “temperature”
   Self-assembly is defined as a relation between
    configurations:




                    A                               B
Example of Tile Systems
Sierpinski tile set
     7 types of tiles: 1 seed, 2 boundary (input) tiles, 4 rule tiles

                                                             Boundary
      Seed
                                                             tiles




                            “Rule” tiles
Sierpinski tile set
                      Tiles
Sierpinski tile set
Binary counter
   Tyle types:
Binary counter
   Begin with seed
                                 Tiles
   Continue with boundary
                             8
    tiles
                             7
   Then “rule” tiles        6
   Count upwards (binary)   5
           =1                4
                             3
           =0
                             2
                             1
Square self-assembly
   Example for 9x9 square
   41 Tile types
More on assemblies
        Tile additions are non-deterministic
    1.    Several locations for adding tiles
    2.    Several possible tiles could be added in one spot


        Defininitions:               From binary counter

         input sides                output
         propagation (output) sides
                                                      terminal
         terminal sides.
                                              input
Final Assembly Theorem
   Definition: an assembly is locally deterministic if:
       every tile addition has strength 2.
       if tile at (i,j) and all tiles touching its propagation sides are
        removed, then there is only one tile type that can be added at (i,j)


   Theorem:
     “If a tile set has one locally determinist assembly sequence, then
          the same final assembly is produced regardless of order of tile
          additions”.
From theory to practice
(biology)
   Tiles are “do-able” in practice
   DNA Nano-technology




                           Winfree, E. et at. Design and self-assembly of two
                           dimensional. DNA crystals. 1998.
More on the technology
   More tiles from DNA




             Hao Yan et al. “4x4 DNA Tile and Lattices: Characterization, Self-
             Assembly and Metallization of a Novel DNA Nanostructure Motif” 2003.
 Road ahead…
1.       First paper (complexity):
         Ties (Kolmogorov) computation of a shape with
          complexity of tile system that self-assembles it
         Note: the former has nothing to do with self-assembly


Robust self-assembly
2.       Second paper (fault-tolerance):
         how to avoid nucleation and growth errors
3.       Third paper (self-healing):
         how to avoid gross damage
First paper
   D. Soloveichik, E.Winfree,
    “Complexity of Self-Assembled Shapes”
Coordinated shapes
   Let S be a finite set of locations in Z2.
    S is a coordinated shape if it’s connected




    Coordinated shape            This is not
Transforming coordinated shapes
   Scalings:




   Translations:
Shapes
   Scale and translation equivalence relations
    on coordinated shapes define class of
    shapes




        coordinated shape 1   coordinated   coordinated shape 3
                                shape 2

         All belong to the same class of shapes         =
Computer Science Concepts
   Kolmogorov complexity:

            = size of the smallest program outputting the
        coordinated shape as a list of locations



       Similarly
Tile Complexity
   The tile-complexity of a coordinated shape S is:
                  n s.t. exists a tile system T of n tile types
      Ksa(S)= min that uniquely produces assembly A and
                  S is the coordinated shape of A.

                  n s.t. exists a tile system T of n tile types
      Ksa( )= min that uniquely produces assembly A and
                    is the shape of A.
Main Theorem
   There exist constants          such that for any shape ,




   To show the right inequality, the paper explicitly shows
    how to find an optimum tile system that assembles a
    given shape! (proof in paper)
   The minimun number of bits required to store n tile types
    is           i.e., same complexity as Kolmogorov
    complexity of shape!
Corollary
       (tile complexity) of shapes is uncomputable.
    i.e., given a shape, the minimun number of tiles required
       to assemble it cannot be computed

   Formally, the following language is undecidable:
Second paper
   H. Chen, A. Goel,
    “Error Free Self-Assembly using Error Prone Tiles”
Kinetic Tile Assembly Model
(kTAM)
   Stochastic model.
   Add and remove tiles
   Kinetics:




   Two parameter:
Snake proof reading
A  simple one dimensional example will be used to
illustrate the algorithm.

We  will consider four tile types, with two bond
types + null bond:
Snake proof reading (2)
   The input will consist of a structure of n+2 tiles
   Our 1-D crystal will output the parity of the input.
     input=“1111” (n=4)




                             0 0       1 1       0 0       1 1       0
                                   1         1         1         1
                                   1         1         1         1
Definition
   Insufficient attachment (at           ) is:
         A process where a tile attaches with strength one but
         before it falls, another tile attaches next to it
         (and now both are held by strength         ).


   They can cause two type of errors:
       growth errors
       nucleation errors.
Growth Errors
   A growth error is an invalid tile attachment to a
    “valid” position


                0 0       1 1       1 1       0 0       1
                      1         0         1         1
                      1         1         1         1




                          Incorrect pairing
Nucleation Errors
   When a tile attaches to an incorrect position
    (small binding strength)


                  0 0       1         1       0 0       1
                        1                 1         1
                        1        1        1         1




        Correct pairing, but weak bonding..that then stabilizes
Winfree-Bekbolatov
proofreading system
   Replace each tile with 2x2 blocks.
   The internal glues are all unique to the 2x2 block




   Corrects for growth but not nucleation errors.
Snake proofreading

   Replace each tile with 2x2 blocks.
   The internal glues are all unique to the 2x2
    block




   Corrects for growth and nucleation errors.
Robust parity check
   Replace tiles with 2x2 blocks
   Note location of strong bonds and null bonds

 0 T 0B        X3 X3        1T 1T        X7 X7        0T 0B        X3 X3        1T 1T        X7 X7        0T
          X2           X4           X6           X8           X2           X4           X6           X8

          X2           X4           X6           X8           X2           X4           X6           X8
 0 B 0B                     1B 1B           X5        0B 0B                     1B 1B                     0B
          1L           1R           1L           1R           1L           1R           1L           1R

          1L           1R           1L           1R           1L           1R           1L           1R
Insufficient attachments

 0 T 0B        X3 X3        1T
                                                       Insufficient Attachment
          X2           X4

          X2           X4                                      X8           X2
 0 B 0B                     1B                            X5        0B 0B
          1L           1R                                      1R           1L

          1L           1R           1L   1R       1L           1R           1L       1R




Continuous                        No          Weak tile               Insufficient
Markov Chain                     error        attaches                attachment
Model
Insufficient attachments
                       No         Insufficient
                      error       attachment




   1 insufficient attachment is very unlikely, but over the
    course of n attachments, the probability of getting at least
    one insufficient attachment might become significant.
   Snake-proofreading requires two insufficient attachments
    in close proximity to have an error than can propagate.
Nucleation error improvement
                                       Cannot propagate with tau=2 unless
  0 T 0B        X3 X3        1T        another insufficient attachment occurs
           X2           X4

           X2           X4                                      X8           X2
  0B 0B                      1B                            X5        0B 0B
           1L           1R                                      1R           1L

           1L           1R        1L     1R           1L        1R           1L   1R




                                            1                     2
                         No            Insufficient          Insufficient
                        error          attachment           attachments
General Snake proofreading
   Previous example only considered one
    directional growth.
   General method extends to:
       L-bounded systems (growth S  N and E  W)
       Replaces a tile by k x k block
   Set of rules to construct internal bonds
       All internal bonds are unique to the tile block
       Most of them have strength 1, some have strength 0,
        some have strength 2.
Example
   Notice how tiles are constructed following a
    snake pattern.
                  T4,1 T4,2 T4,3 T4,4

                  T3,1 T3,2 T3,3 T3,4

                  T2,1 T2,2 T2,3 T2,4

                  T1,1 T1,2 T1,3 T1,4
Main analytical results
   Theorem 1
      With a 2k x 2k snaked tile system (for k sufficiently large) assuming
      we can set       to be    , an N x N square of blocks can be
      assembled in time        and w.h.p no block errors happen for
      time after that.

   Theorem 2
      With a 2k x 2k snaked tile system (          ) assuming that we can
      set    to be     , an N x N square of blocks can be assembled in
      time      and w.h.p no block errors happen for     time after that.

    Informally, snaked proofreading results in tile systems
    which assemble quickly and remain stable for long time
Simulation results (1)
   Three systems simulated using xgrow
       no proofreading, WB proofreading, snaked
   4x4 tile blocks
Simulation results (2)
   Simulations relaxed idealized modeling conditions
   They corroborate analytical results
Third paper
   E. Winfree,
    “Self-Healing Tile Sets”
    A new type of error:
    Gross damage
   Gross damage: removes a region containing many tiles
   This error is rare in kTAM model, but is not hard to
    imagine in practice
       ripping induced by fluid flow
       interactions with other objects in solution)
   Is healing possible? Under general assumptions, Yes!
   We formulate the answer in the framework of aTAM:
        “focus on the information-propagation aspects of the problem
        rather than on the probabilistic aspects”
Self-healing tiles
   Definition:
      A tile system is self-healing (in the aTAM) if,
      for any produced assembly, the following holds:

      “If n tiles are removed such that all remaining tiles are
      still connected to the seed tile, then subsequent growth
      is guaranteed to eventual restore every removed tile
      without error in time O(n)”
    Are self-assembled patterns
    self-healing?
   Is any of our previous examples self-healing? NO!
   Example:




                                   BANG!
Main results
   3x3 block transformation:
       repair tile sets that grow in a quarter-plane from L-
        shaped boundary
   5x5 transformation:
       more general case, work for our three examples
   7x7 transformation:
       even more general, works for “polyomino” tile sets.
3x3 Transformations
   L-bounded tile sets are self-healing under
    these transformation
   Quarter plane growth from L-shape boundary
    is a rich class of tile sets
       capable of creating a variety of patterns
       sufficient for universal computation
3x3 Transformations
   For every tile type, we introduce 9 new tile types
   Define tile-type bonds, and bond-type bonds




   The 3x3 block transformation shown above produces a
    self-healing tile set when applied to an L-bounded tile set.
3x3 Transformations
   Example:
5x5 Transformation Theorem
Definition:
   A transformable tile set is a locally
   deterministic tile set such that
  1.   each tile type always appears with the same
       sides as input, propagation, and terminal sides
  2.   Termina sides have null bonds.
5x5 Transformation Theorem
Theorem:
   The 5x5 block transformation shown below produces a
    self-healing tile set when applied to a transformable tile
    set.
7x7 Transformation Theorem
   Problem: Some of the new tiles can act as “seeds”.

   Definition: A polyomino set is a tile set that also
    includes block of tiles (polyominos) that can grow
    from tiles that are not original seed tiles.

   Theorem:
    An 7x7 block transformation can produce a self-
    healing tile set, even with a polyomino set.
7x7 Transformation Theorem
Transformations:
Open questions
There are still other type of errors that can occur
    e.g., when seed is removed, or continual gross damage
   Can we combine in a single method or
    transformation, a scheme that is robust under a
    wide class of type of errors and tile types?
   Are there smaller self-healing tile sets?
   What about errors that damage a tile itself?

   What about 3D assemblies (theory and practice)?
THE END


 Thanks!

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