Robust Self-Assembly of DNA

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Robust Self-Assembly of DNA

• Eduardo Abeliuk
• Dept. of Electrical Engineering
• Stanford University
• November 30, 2006

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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

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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.
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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
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Rothemund PWK, Folding DNA to create nanoscale
shapes and patterns, Nature 2006
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Rothemund PWK, Folding DNA to create nanoscale
shapes and patterns, Nature 2006
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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.
• The assembly of a 2D crystal can simulate a
  universal Turing machine!

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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
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The tile assembly model

• Infinite lattice
• Z x Z
• Every position in
• the grid has a relative
• position associated
• N(i,j)(i,j1)
• S(i,j)(i,j-1)
• E(i,j)(i1,j)
• W(i,j)(i-1,j)
• 
  (i,j)

N
W
E
S
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Bond types and Tile Types

• Our fundamental unit is a square tile with
  labelled edges, or bond types.
• 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.
• We will represent tile types with different
  colors. All tile types for the set T.

A B C D
D A C D
B A B D
A B C D
B A B D
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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 2
Configuration 1
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Interaction between tiles

• A strength function
• defines the interactions between two tiles.
• We say a tile t1 interacts with its neighbor t2
  with strength
• Usually, only diagonal strength functions are
  considered, and the range of g is 0,1,2

g A B C D null A 1 0 0 0 0 B
0 1 0 0 0 C 0 0 2 0 0 D
0 0 0 1 0 null 0 0 0 0 0
A B C D
B A B C
A C C D
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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
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Example of Tile Systems

• Sierpinski tile set
• 7 types of tiles 1 seed, 2 boundary (input)
  tiles, 4 rule tiles

Boundary tiles
Seed
Rule tiles
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Sierpinski tile set
Tiles
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Sierpinski tile set
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Binary counter

• Tyle types

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Binary counter

• Begin with seed
• Continue with boundary
• tiles
• Then rule tiles
• Count upwards (binary)
• 1
• 0

Tiles
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Square self-assembly

• Example for 9x9 square
• 41 Tile types

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More on assemblies

• Tile additions are non-deterministic
• Several locations for adding tiles
• Several possible tiles could be added in one spot
• Defininitions
• input sides
• propagation (output) sides
• terminal sides.

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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.

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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.
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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.
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Road ahead

• 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
• Second paper (fault-tolerance)
• how to avoid nucleation and growth errors
• Third paper (self-healing)
• how to avoid gross damage

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First paper

• D. Soloveichik, E.Winfree,
• Complexity of Self-Assembled Shapes

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Coordinated shapes

• Let S be a finite set of locations in Z2.
• S is a coordinated shape if its connected

Coordinated shape
This is not
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Transforming coordinated shapes

• Scalings
• Translations

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Shapes

• Scale and translation equivalence relations on
  coordinated shapes define class of shapes

coordinated shape 1
coordinated shape 3
coordinated shape 2
All belong to the same class of shapes
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Computer Science Concepts

• Kolmogorov complexity
• size of the smallest program outputting
  the coordinated shape as a list of locations
• Similarly

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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.

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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!

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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

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Second paper

• H. Chen, A. Goel,
• Error Free Self-Assembly using Error Prone
  Tiles

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Kinetic Tile Assembly Model (kTAM)

• Stochastic model.
• Add and remove tiles
• Kinetics
• Two parameter

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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

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Snake proof reading (2)

• The input will consist of a structure of n2
  tiles
• Our 1-D crystal will output the parity of the
  input.
• input1111 (n4)

0
0 1 1
1 0 1
0 1 1
1 0 1
1
1
1
1
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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.

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Growth Errors

• A growth error is an invalid tile attachment to a
  valid position

0
0 1 1
1 1 0
1 0 1
0 1 1
1
1
1
1
Incorrect pairing
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Nucleation Errors

• When a tile attaches to an incorrect position
  (small binding strength)

0
0 1 1
1 0 1
0 1 1
1
1
1
1
Correct pairing, but weak bonding..that then
stabilizes
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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.

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Snake proofreading

• Replace each tile with 2x2 blocks.
• The internal glues are all unique to the 2x2
  block
• Corrects for growth and nucleation errors.

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Robust parity check

• Replace tiles with 2x2 blocks
• Note location of strong bonds and null bonds

0T
0B X3 X2
X3 1T X4
X3 1T X4
0B X3 X2
1T X7 X6
X7 0T X8
1T X7 X6
X7 0T X8
0B
X2 0B 1L
X2 0B 1L
X4 1B 1R
X4 1B 1R
X8 X5 0B 1R
X6 1B 1L
X6 1B 1L
X8 0B 1R
1L
1L
1R
1L
1R
1R
1R
1L
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Insufficient attachments
0T
0B X3 X2
X3 1T X4
0B
X2 0B 1L
X8 X5 0B 1R
X4 1B 1R
X2 0B 1L
1L
1L
1R
1L
1R
1R
1R
1L
Continuous Markov Chain Model
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Insufficient attachments

• 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.

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Nucleation error improvement
Cannot propagate with tau2 unless another
insufficient attachment occurs
0T
0B X3 X2
X3 1T X4
0B
X2 0B 1L
X8 X5 0B 1R
X4 1B 1R
X2 0B 1L
1L
1L
1R
1L
1R
1R
1R
1L
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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.

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Example

• Notice how tiles are constructed following a
  snake pattern.

T4,2
T4,1
T4,3
T4,4
T3,1
T3,3
T3,2
T3,4
T2,1
T2,2
T2,3
T2,4
T1,1
T1,2
T1,4
T1,3
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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

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Simulation results (1)

• Three systems simulated using xgrow
• no proofreading, WB proofreading, snaked
• 4x4 tile blocks

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Simulation results (2)

• Simulations relaxed idealized modeling conditions
• They corroborate analytical results

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Third paper

• E. Winfree,
• Self-Healing Tile Sets

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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

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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)

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Are self-assembled patterns self-healing?

• Is any of our previous examples self-healing? NO!
• Example

BANG!
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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.

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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

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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.

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3x3 Transformations

• Example

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5x5 Transformation Theorem

• Definition
• A transformable tile set is a locally
  deterministic tile set such that
• each tile type always appears with the same sides
  as input, propagation, and terminal sides
• Termina sides have null bonds.

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5x5 Transformation Theorem

• Theorem
• The 5x5 block transformation shown below produces
  a self-healing tile set when applied to a
  transformable tile set.

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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.

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7x7 Transformation Theorem

• Transformations

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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)?

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THE END

• Thanks!
• Questions?