1. Matthew J. Patitz Explorations of Theory and Programming in Self-Assembly Matthew J. Patitz Department of Computer Science University of Texas-Pan American 10/19/2010

2. Matthew J. Patitz Introduction to self-assembly A: The process by which relatively simple components in a disorganized state autonomously combine to form more complex objects Q: What is self-assembly?

3. Matthew J. Patitz Introduction to self-assembly Q: What self-assembles? A: Many structures at many scales! Examples include: Crystals (such as snowflakes) Biological structures (e.g. viruses) Cosmic structures (e.g. galaxies) Images courtesy of SnowCrystals.com Image courtesy of NSF.govImage courtesy of hubblesite.org A: The process by which relatively simple components in a disorganized state autonomously combine to form more complex objects Q: What is self-assembly?

4. Matthew J. Patitz Introduction to self-assembly Q: What self-assembles? A: Many structures at many scales! Examples include: Crystals (such as snowflakes) Biological structures (e.g. viruses) Galaxies A: The process by which relatively simple components in a disorganized state autonomously combine to form more complex objects Q: What is self-assembly? Q: Why study self-assembly? A: Several reasons: Better understand origin and functioning of living systems Mathematically interesting properties Eventual creation of ‘fantastic’ technologies

5. Matthew J. Patitz Directions in self-assembly research Toward atomically-precise manufacturing IBM-Caltech collaboration to use self-assembled molecules to guide design of smaller processors Credit: PRNewsFoto/IBM Nano biomedical devices Aarhus University Center for DNA Nanotechnology’s box with programmable lid Credit: : Ebbe S. Andersen, Aarhus University

6. Matthew J. Patitz Researchers study both natural and artificial self-assembling systems Theoretical as well as experimental work My focus is on theoretical research into an artificial model, which I will now introduce… Directions in self-assembly research

7. Matthew J. Patitz Tile Assembly Model Erik Winfree introduced the Tile Assembly Model (TAM) in 1998 It was later refined by Paul Rothemund The TAM is based on experimental work with DNA molecules by Ned Seeman * Both Winfree and Rothemund have both been named MacArthur Fellows

8. Matthew J. Patitz Tile Assembly Model DNA molecules formed into shapes such as Holliday junctions can be treated logically as 2-dimensional squares Molecular structure of a Holliday junction (Image courtesy of Wikipedia) Schematic view of Holliday junction with extended ‘sticky ends’ With the ‘sticky ends’ treated as glues, these molecules can be thought of as square ‘tiles’

9. Matthew J. Patitz Tile Assembly Model Fundamental components are 2-D square tiles Each side has an associated glue, with: A type (usually represented by a string value) An integer-valued strength (usually 0, 1, or 2) Tiles can also have labels (non- functional, for convenience) Tiles cannot be rotated Finite number of different tile types An infinite supply of each tile type Abutting sides of tiles bind if both glue strengths and values match Those sides bind with that shared strength A tile can bind to an assembly if the sum of binding strengths is at least equal to the “temperature” value of the system (usually 1 or 2) Assembly begins from a “seed” tile or assembly and grows 1 tile at a time Strength 0 Strength 1 Strength 2

10. Matthew J. Patitz Temperature value = 2 Seed = (S, (0,0)) Tile assembly example Strength 0 Strength 1 Strength 2 Tile set:

11. Matthew J. Patitz Tile assembly example Attachment by 2 strength-1 bonds is a form of “cooperation” between multiple tiles that gives the model great power 5432154321

12. Matthew J. Patitz Tile Assembly Model A tile assembly system (TAS) is an ordered triple T =(T,σ,  ) T is the tile set (a set of tile types) σ defines the seed assembly (tile types and locations)  is an integer value specifying the temperature (the minimum total binding strength required for a tile to adhere to an assembly) A TAS is directed if it has a single, unique final assembly

13. Matthew J. Patitz Self-assembly of shapes Any finite shape can trivially be self-assembled by creating a hard-coded tile type for every position in the shape. To test the theoretical limits of the TAM, we explore infinite shapes Self-similar fractals are interesting infinite shapes because of their complex, aperiodic nature

14. Matthew J. Patitz A (non-trivial) discrete self-similar fractal is a recursively defined, infinite set of integer lattice points having fractal dimension more than 1 but less than 2. c The second stage is the generator of the fractal. Discrete self-similar fractals

15. Matthew J. Patitz Example Discrete Self-Similar Fractal: The Sierpinski Carpet

16. Matthew J. Patitz Self-assembly of discrete self-similar fractals In Self-Assembly of Discrete Self-Similar Fractals, Patitz and Summers showed that there are classes of discrete self-similar fractals that don’t self-assemble in the TAM We [Patitz and Summers] also proved that for an overlapping class, there are approximations that do self-assemble

17. Matthew J. Patitz Self-assembly of discrete self-similar fractals A pinch point discrete self-similar fractal is a discrete self-similar fractal having a “special” kind of generator. The generator must be connected.

18. Matthew J. Patitz Self-assembly of discrete self-similar fractals Theorem. No pinch point discrete self-similar fractal strictly self-assembles in a directed tile assembly system (at any temperature). Why? Because of the geometry of pinch point discrete self-similar fractals. Question. Do any non-trivial discrete self-similar fractals strictly self-assemble?

19. Matthew J. Patitz Self-assembly of discrete self-similar fractals A nice discrete self-similar fractal is any discrete self- similar fractal whose generator looks like this… But NOT these…

20. Matthew J. Patitz Self-assembly of discrete self-similar fractals For any nice self-similar fractal, we can apply “fiber” to it as follows.

21. Matthew J. Patitz Self-assembly of discrete self-similar fractals Start with the third stage of any nice self-similar fractal.

22. Matthew J. Patitz Self-assembly of discrete self-similar fractals Add some fiber.

23. Matthew J. Patitz Self-assembly of discrete self-similar fractals Recursively build the next stage.

24. Matthew J. Patitz Self-assembly of discrete self-similar fractals And repeat!

25. Matthew J. Patitz Fibered Sierpinski carpet

26. Matthew J. Patitz Theorem. Every nice self-similar fractal has a fibered version that strictly self-assembles and has the same fractal dimension as its non-fibered counter-part. [Patitz and Summers, 2008] Self-assembly of discrete self-similar fractals On to programming tools now…

27. Matthew J. Patitz Simulation of Self-Assembly in the Abstract Tile Assembly Model with ISU TAS Matthew J. Patitz

28. (Iowa State University Tile Assembly Simulator) Overview of ISU TAS Open source C++ application based on wxWidgets Cross platform (Windows, Linux, Mac) Graphical tile set editor Simulator Many debugging features Supports several variations of the model

29. Matthew J. Patitz Tile set editor

30. Matthew J. Patitz Provides a simple graphical representation of the tile set (separate from simulator) Allows creation of new tiles and editing of existing tiles Functionality for copying, pasting, rotating, searching, etc. Displays tile set information such as which tiles are functional duplicates of each other, which tiles are used in the current assembly, etc. Tile set editor

31. Matthew J. Patitz Simulator

32. Matthew J. Patitz 3-D

33. Matthew J. Patitz Simulations of assemblies begin from a user-defined seed Simulations can proceed in single steps or in fast-forward mode Steps are cached, so simulations can also be ‘rewound’ User can set arbitrary zoom factors An overview window shows the entire assembly Frontier locations can be highlighted and toggled Simulator

34. Matthew J. Patitz Downloading the software ISU TAS, the fractal generator, and other related software is all freely available (both the executables and the source code) from the following web site: http://www.cs.iastate.edu/~lnsa

35. Matthew J. Patitz Roadmap for the future Many open problems in the TAM yet to explore Collaborations will be key Fractals, temperature 1, computations vs. space requirements, fault tolerance, etc. Moving beyond the TAM It’s an elegant and powerful model Extremely basic – doesn’t reflect complexity of biological systems Create new, more powerful models Examples: 2-handed assembly DNA/RNA tiles model “Reusable” space, greater impact of geometry Dynamic, adaptable components Emphasis on experimental practicality Theory, programming tools, and laboratory experimentation Inter-disciplinary research – let’s implement them in the lab!

36. Matthew J. Patitz Thank you!