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

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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.

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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]

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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]

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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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

)

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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.

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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.

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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)

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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

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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

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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.

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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.

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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.

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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*).

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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)

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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.

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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)