DNA SelfAssembly For Constructing 3D Boxes

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DNA Self-AssemblyFor Constructing 3D Boxes

• Ming-Yang Kao Vijay RamachandranNorthwestern
  University Yale UniversityEvanston, IL, USA New
  Haven, CT, USA

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Self-Assembly and Nanotechnology

• DNA Tile Self-Assembly
• Goal Perform computations using local rules
  governing how tiles fit together.
• Tiles are made from DNA. Watson-Crick
  hybridization causes exposed bases on certain
  tiles to bind.

• DNA Nanotechnology
• Goal Build small structures with high precision.
• Molecular units are made of DNA and can have
  different shapes.
• 3D structures have been created, but they are not
  scalable.

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Previous Work

• DNA Tile Self-Assembly
• Theory of tilingWang 61
• Model for 2D DX computationWinfree 95
• TX computationLaBean, Winfree,and Reif 99

• DNA Nanotechnology
• Development of DNA subunits Seeman 82
• DX moleculesFu and Seeman 93
• TX moleculesLaBean et al. 00
• 3D CubeChen andSeeman 91

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Combining Two Technologies

• Use the well-studied properties of tile
  self-assembly to create a model for nanostructure
  fabrication.
• Objects consist of DNA tiles synthesized to fit
  together like puzzle pieces.
• Self-assembly of DX molecules to build 2D
  lattices of DNA Winfree et al. 98
• 2D mathematical model and complexity measure
  Rothemund and Winfree 00

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Extending the Model to 3D

• A natural extension of RW 00 is the creation
  of 3D structures by tiling.
• Problem 1 What are the natural molecular
  building blocks?
• Problem 2 How do we retain the scalability of 2D
  nanostructure fabrication?
• Our approach consider the (most interesting)
  case of using 2D tiles to build 3D structures.

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Objective

• Develop a model for constructing 3D
  nanostructures using 2D tiles.
• Support different structures of different sizes.
• Closely match the behavior of tiles in solution.
• Develop algorithms to build a hollow cube.
• Analyze these algorithms theoretical properties
  and biological feasibility using appropriate
  complexity measures.

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Basic Idea

• Use 2D tiles to form a planar shape that can fold
  into a box.
• When corresponding edges are in proximity, the
  exposed bases should attract each other and cause
  slow folding.

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The Need For Randomization

• Self-assembly requires many copies of all tile
  types.
• Traditional 2D self-assembly is deterministic
  tiles form a predictable pattern.
• What happens when shapes interfere with each
  other?
• Prevent this by making each shape unique start
  each with randomized seed tiles.

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Another Issue

• Although edges on different shapes need to be
  different, certain edges within the same shape
  must correspond.
• This paper formalizescopy patterns to shift the
  random information from seed tiles to the edges.
• Implementation details yield different
  complexities.

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Our Model Molecular Level

• Use tRNA-style molecules (c), or
  branched-junction molecules (b) Seeman 82.
• Truly four-faced, unlike DX or TX molecules (a)
• Stable backbone, though flexible enoughto align
  properly for folding

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Our Model Symbolic Level

• DNA sequence s of length n5-b1b2...bn-3,
  where bi? A,C,T,G
• Watson-Crick complementations
  3-b1b2...bn-5 AT, CG (s) s
• The concatenation of ss1...sn andtt1...tm is
  sts1...snt1...tm
• The subsequence of ss1...sn from i to j is si
  jsisi1..sj-1sj

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Our Model Symbolic Level

• Hybridization occurs between two strands with
  complementary subsequences. Assume no
  misbindings.
• Threshold temperature the solution temperature
  above which a double-stranded DNA molecule
  denatures. Formally, some ??T such that the
  strand denatures in solution of temperature above
  ???(??,?-?) for ?gt0.

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DNA Tiles

• Let W be a set of DNA words and S be a set of
  symbols. Define an encoding map enc S ? W.
• A DNA tile is a 4-tuple of symbols(sN, sE, sS,
  sW) where si?S and enc(si) is the exposed
  sequence on the action site.

sN
5
enc(sE)
sE
3
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k-Level Generalizations

• Some algorithms require more flexibility than in
  the one-word-per-side model.
• Solution allow each side to be a k-tuple from a
  symbol set ?k. Let each tuple correspond to a
  DNA sequence using a map similar to enc.
• The concatenation generalization concatenates the
  words encoding the symbols in ? on the side of a
  tile.

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Algorithmic Procedures

• One step consists of
• Adding tiles to solution.
• Deterministic rule only one tile type fits in a
  given position.
• Randomized rule several tile types could fit in
  a given position probability is proportional to
  the concentration of tiles added.
• Waiting for tiles to hybridize, cycling
  temperature to prevent or induce binding.
• Washing away excess, if necessary.

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Complexity Measures

• Time complexity number of steps
• Space complexity number of tile types
• Alphabet size number of words
• Temperatures number of threshold temperatures
  needed
• Generalization level how much information per
  tile side (how many words per side, or size of
  tiles in base-pairs)
• Misformation probability probability that at
  some step, a tile binds incompletely (not on all
  the sides it should)

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Hollow Cube Algorithms

• 3-level generalizations.
• Define a set of words? ?1,?2,,?p, used
  toform random sequences.
• From randomized seed tiles (e.g., base strip),
  copy the pattern to edges (using shaded regions,
  except for edges at A and D).
• Cut away shaded region by increasing temperature.
  The remaining tiles can then fold.

?3
?1
?4
?7

G
H
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Assembly and Copy Patterns

• Random Assembly used to build the randomized
  seed tiles
• Straight Copy used to copy an exposed sequence
  through to a parallel end of an adjacent region
  (deterministic)
• Turn Copy used to copy an exposed sequence to a
  perpendicular end of an adjacent region
  (deterministic)

Straight Copy
Turn Copy
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Row-By-Row Algorithm

• Randomized assembly is used exactly where needed
  on the shape. The edge is then copied to its
  corresponding location.
• Straight copy is performed one row per step.
  Only one counter (current row) is needed, and
  temperature-sensitive binding is used to prevent
  misformations (?i are the strongest).
• Turn copy is performed with horizontal and
  vertical counters on the tiles. Tiles along the
  diagonal shift the DNA sequence.

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Row-By-Row Analysis

• n length of a cube edge (in tiles)
• p number of patterns. Then
• Alphabet size is 8n p O(1).
• Time complexity is 5n O(1).
• Space complexity is6n2p 10np 4p 8n O(1).
• The number of distinct temperatures required is
  3.
• Misformation probability is 0.

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All-Together Algorithm

• Random assembly is performed before copy steps
  for one of each pair of corresponding edges.
  Each strip is marked with position counters so it
  binds at the correct location.
• Straight copy and turn copy are done in one step.
  Every tile has a horizontal and vertical counter
  and a pattern in ?, so it should fit in exactly
  one spot.

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All-Together Analysis

• n length of a cube edge (in tiles)
• p number of patterns. Then
• Alphabet size is 8n p O(1).
• Time complexity is O(1).
• Space complexity is 16n2p O(1).
• The number of distinct temperatures required is 2
  (3).
• Misformation probability is 1-(1/pn) (0).

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Other Algorithms (?)

• By-Region remove most counters by controlling
  growth in only certain rows and columns of a
  region. (High misformation probability)
• Border-first construct the frame of regions
  first, and then fill in the structure with
  generic tiles containing no information.
  (Stability problems)
• Build faces separately, or split folding by
  building sets of three faces together. (Cannot
  guarantee that sides eventually match and the
  cube forms in solution)

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Summary of Contributions

• Developed an abstract model of self-assembly that
  closely models the behavior of DNA tiles
• Allows construction of scalable 2D and 3D
  nanostructures
• Formalizes use of temperature and DNA words
• Provides several measures for analysis
• Identified and solved problems central to
  building 3D structures from 2D tiles by
  introducing assembly and copy patterns, including
  randomization
• Explored and analyzed several algorithms for
  building a hollow cube.

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Possibilities for Further Work

• Improve algorithms by reducing number of tiles,
  number of steps, or both.
• Is less information necessary? (2- or 1-level
  generalizations, or fewer randomized seed tiles)
• Develop or use stronger molecular unitsor
  proteins to help the folding process
• New algorithms for other structures (possibly
  with important biochemical uses)