DNA Self-Assembly For Constructing 3D Boxes

1. DNA Self-AssemblyFor Constructing 3D Boxes Ming-Yang Kao Vijay RamachandranNorthwestern University Yale UniversityEvanston, IL, USA New Haven, CT, USA

2. 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. Self-Assembly and Nanotechnology DNA Self-Assembly For Constructing 3D Boxes

3. 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] Previous Work DNA Self-Assembly For Constructing 3D Boxes

4. 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] DNA Self-Assembly For Constructing 3D Boxes

5. 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. DNA Self-Assembly For Constructing 3D Boxes

6. 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. DNA Self-Assembly For Constructing 3D Boxes

7. 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. DNA Self-Assembly For Constructing 3D Boxes

8. 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. The Need For Randomization DNA Self-Assembly For Constructing 3D Boxes

9. 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. Another Issue DNA Self-Assembly For Constructing 3D Boxes

10. 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 DNA Self-Assembly For Constructing 3D Boxes

11. Our 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 DNA Self-Assembly For Constructing 3D Boxes

12. 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 >0. DNA Self-Assembly For Constructing 3D Boxes

13. 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’ DNA Self-Assembly For Constructing 3D Boxes

14. 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. DNA Self-Assembly For Constructing 3D Boxes

15. 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. DNA Self-Assembly For Constructing 3D Boxes

16. 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) DNA Self-Assembly For Constructing 3D Boxes

17. 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. Hollow Cube Algorithms 3 4 1 7 … G H DNA Self-Assembly For Constructing 3D Boxes

18. 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 DNA Self-Assembly For Constructing 3D Boxes

19. 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. DNA Self-Assembly For Constructing 3D Boxes

20. 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. DNA Self-Assembly For Constructing 3D Boxes

21. 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. DNA Self-Assembly For Constructing 3D Boxes

22. 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*). DNA Self-Assembly For Constructing 3D Boxes

23. 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) DNA Self-Assembly For Constructing 3D Boxes

24. 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. DNA Self-Assembly For Constructing 3D Boxes

25. 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) DNA Self-Assembly For Constructing 3D Boxes