Explorations of Theory and Programming in Self-Assembly

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

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

3. Introduction to self-assembly Q: What is self-assembly? Q: What self-assembles? A: The process by which relatively simple components in a disorganized state autonomously combine to form more complex objects • 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.gov Image courtesy of hubblesite.org Matthew J. Patitz

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

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

6. Directions in self-assembly research • 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… Matthew J. Patitz

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

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

9. Strength 0 Strength 1 Strength 2 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 Matthew J. Patitz

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

11. Tile assembly example 5 4 3 2 1 Attachment by 2 strength-1 bonds is a form of “cooperation” between multiple tiles that gives the model great power Matthew J. Patitz

12. Tile Assembly Model • A tile assembly system (TAS) is an ordered triple T=(T,σ,t) • T is the tile set (a set of tile types) • σdefines the seed assembly (tile types and locations) • t 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 Matthew J. Patitz

13. 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 Matthew J. Patitz

14. Discrete self-similar fractals • 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. • The second stage is the generator of the fractal. c Matthew J. Patitz

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

16. 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 Matthew J. Patitz

17. 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. Matthew J. Patitz

18. 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? Matthew J. Patitz

19. 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… Matthew J. Patitz

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

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

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

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

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

25. Fibered Sierpinski carpet Matthew J. Patitz

26. Self-assembly of discrete self-similar fractals • 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] On to programming tools now… Matthew J. Patitz

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

28. Overview of ISU TAS (Iowa State University Tile Assembly Simulator) • 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 Matthew J. Patitz

29. Tile set editor Matthew J. Patitz

30. Tile set editor • 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. Matthew J. Patitz

31. Simulator Matthew J. Patitz

32. 3-D Matthew J. Patitz

33. Simulator • 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 Matthew J. Patitz

34. 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 Matthew J. Patitz

35. 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! Matthew J. Patitz

36. Thank you! Matthew J. Patitz