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DNA Self-AssemblyDNA Self-AssemblyFor Constructing 3D For Constructing 3D BoxesBoxes
Ming-Yang Kao Vijay RamachandranNorthwestern University Yale UniversityEvanston, IL, USA New Haven, CT, USA
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 2
Self-Assembly and Self-Assembly and NanotechnologyNanotechnology
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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 3
Previous WorkPrevious Work
DNA Tile Self-Assembly
• Theory of tiling[Wang ’61]
• Model for 2D DX computation[Winfree ’95]
• TX computation[LaBean, Winfree,and Reif ’99]
DNA Nanotechnology• Development of DNA
subunits [Seeman ’82]
• DX molecules[Fu and Seeman ’93]
• TX molecules[LaBean et al. ’00]
• 3D Cube[Chen andSeeman ’91]
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 4
Combining Two Combining Two TechnologiesTechnologies
• 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]
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 5
Extending the Model to 3DExtending 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 6
ObjectiveObjective
• 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 7
Basic IdeaBasic 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 8
The Need For The Need For RandomizationRandomization
• 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 9
Another IssueAnother 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 10
Our Model: Molecular Our Model: Molecular LevelLevel
• 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 enough
to align properly for folding
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 11
Our Model: Symbolic LevelOur Model: Symbolic Level
• DNA sequence s of length n:5’-b1b2...bn-3’, where bi {A,C,T,G}
• Watson-Crick complementation:s = 3’-b1b2...bn-5’; A=T, C=G; (s) = s
• The concatenation of s=s1...sn andt=t1...tm is st=s1...snt1...tm
• The subsequence of s=s1...sn from i to j is s[i : j]=sisi+1..sj-1sj
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 12
Our Model: Symbolic LevelOur 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 >0.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 13
DNA TilesDNA 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 siS and enc(si) is the exposed sequence on the action site. sN
sE
3’
5’
enc(sE
)
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 14
kk-Level Generalizations-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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 15
Algorithmic ProceduresAlgorithmic 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 16
Complexity MeasuresComplexity 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)
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 17
Hollow Cube AlgorithmsHollow 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.
G H
…3 14 7
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 18
Assembly and Copy Assembly and Copy PatternsPatterns
• 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
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 19
Row-By-Row: AlgorithmRow-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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 20
Row-By-Row: AnalysisRow-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 is
6n2p + 10np + 4p + 8n + O(1).• The number of distinct temperatures
required is 3.• Misformation probability is 0.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 21
All-Together: AlgorithmAll-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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 22
All-Together: AnalysisAll-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*).
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 23
Other Algorithms (?)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)
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 24
Summary of ContributionsSummary 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.
10/2/2001 DNA Self-Assembly For Constructing 3D Boxes 25
Possibilities for Further Possibilities for Further WorkWork
• 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)