1. Molecular Algorithms -- Overview
Ashish Goel Stanford University
Joint work with Len Adleman, Holin Chen, Qi Cheng, Ming-Deh Huang, Chris Luhrs, Pablo Moisset, Paul Rothemund, Rebecca Schulman, Erik Winfree 1 1 1 1 1 1 1 1 1 1
Counter made by self-assembly
[Rothemund, Winfree ’00] [Adleman, Cheng, Goel, Huang ’01] [Cheng, Goel, Moisset ‘04]

2. Ashish Goel, ashishg@stanford.edu
Adleman’s Experiment
Use DNA strands to code information
Can solve Hamiltonian paths
NP complete
Lots of press
Ingenious experiment
Started an entire community (eg. The DNA computing workshop)
Not practical, as an alternative to modern computers
Today: Brief overview
Molecular self-assembly
Molecular machines
People/Funding/Current status
Short term and long term directions
Ashish Goel,

3. Molecular Self-assembly
Self-assembly is the spontaneous formation of a complex by small (molecular) components under simple combination rules
Geometry, dynamics, combinatorics are all important
Inorganic: Crystals, supramolecules
Organic: Proteins, DNA
Goals: Understand self-assembly, design self-assembling systems
A key problem in nano-technology, molecular robotics, molecular computation
Ashish Goel,

4. Ashish Goel, ashishg@stanford.edu
A Matter of Scale
Question: Why an algorithmic study of “molecular” self-assembly specifically?
Answer: The scale changes everything
Consider assembling micro-level (or larger) components, eg. robot swarms. Can attach rudimentary computers, motors, radios to these structures.
Can now implement an intelligent distributed algorithm.
In molecular self-assembly, we have nano-scale components. No computers. No radios. No antennas.
Need local rules such as “attach to another component if it has a complementary DNA strand”
Self-assembly at larger scales is interesting, but is more a sub-discipline of distributed algorithms, artificial intelligence etc.
Ashish Goel,

5. The Tile Model of Self-Assembly
[Wang ’61]
Ashish Goel,

6. Synthesized Tile Systems – I
Styrene molecules attaching to a Silicon substrate
Coat Silicon substrate with Hydrogen
Remove one Hydrogen atom and bombard with Styrene molecules
One Styrene molecule attaches, removes another Hydrogen atom, resulting in a chain
Suggested use: Self-assembled molecular wiring on electronic circuits
[Wolkow et al. ’00]
Ashish Goel,

7. Synthesized Tile Systems - II
A DNA “rug” assembled using DNA “tiles”
The rug is roughly 500 nm wide, and is assembled using DNA tiles roughly 12nm by 4nm (false colored)
(Due to Erik Winfree, Caltech)
Ashish Goel,

8. Rothemund’s DNA Origami
A self-folded virus!!
Ashish Goel,

9. Ashish Goel, ashishg@stanford.edu
Abstract Tile Systems
Tile: the four glues and their strengths
Tile System:
K tiles
Infinitely many copies available of each tile
Temperature t
Accretion Model:
Assembly starts with a single seed tile, and proceeds by repeated addition of single tiles e.g. Crystal growth
Are interested primarily in tile systems that assemble into a unique terminal structure
[Rothemund and Winfree ‘00] [Wang ‘61]
Ashish Goel,

10. Is Self-Assembly Just Crystallization?
Crystals do not grow into unique terminal structures
A sugar crystal does not grow to precisely 20nm
Crystals are typically made up of a small number of different types of components
Two types of proteins; a single Carbon molecule
Crystals have regular patterns
Computer circuits, which we would like to self-assemble, don’t
Molecular Self-assembly = combinatorics + crystallization
Can count, make interesting patterns
Nature doesn’t count too well, so molecular self-assembly is a genuinely new engineering paradigm. Think engines. Think semiconductors.
Ashish Goel,

11. DNA and Algorithmic Self-Assembly
We will tacitly assume that the tiles are made of DNA strands woven together, and that the glues are really free DNA strands
DNA is combinatorial, i.e., the functionality of DNA is determined largely by the sequence of ACTG bases. Can ignore geometry to a first order.
Trying to “count” using proteins would be hell
Proof-of-concept from nature: DNA strands can attach to “combinatorially” matching sequences
DNA tiles have been constructed in the lab, and DNA computation has been demonstrated
Can simulate arbitrary tile systems, so we do not lose any theoretical generality, but we get a concrete grounding in the real world
The correct size (in the nano-range)
Ashish Goel,

12. A Roadmap for Algorithmic Self-Assembly
Self-assembly as a combinatorial process
The computational power of self-assembly
Self-assembling interesting shapes and patterns, efficiently
Automating the design process?
Analysis of program size and assembly time
Self-assembly as a chemical reaction
Entropy, Equilibria, and Error Rates
Reversibility
Connections to experiments
Self-assembly as a machine
Not just assemble something, but perform work
Much less understood than the first three
Ashish Goel,

13. Can we create efficient counters?
Ashish Goel,

14. Can we create efficient counters?
Yes! Eg. Using “Chinese remaindering”
T=2
Ashish Goel,

15. Can we create efficient counters?
Yes! Eg. Using “Chinese remaindering”
T=2
Ashish Goel,

16. Can we create efficient counters?
Yes! Eg. Using “Chinese remaindering”
T=2
Ashish Goel,

17. Can we create efficient counters?
Yes! Eg. Using “Chinese remaindering”
T=2
Ashish Goel,

18. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
Ashish Goel,

19. Can we create efficient counters?
Yes! Eg. Using “Chinese remaindering”
T=2
Generalizing:
Say p1, p2, …, pk
are distinct primes.
We can use i pi tiles
to assemble a
k £ (i pi) rectangle.
Ashish Goel,

20. Ashish Goel, ashishg@stanford.edu
Also, binary
From Cheng, Moisset, Goel;
Optimal self-assembly of counters
at temperature 2
(basic ideas developed over many papers)
Ashish Goel,

21. Ashish Goel, ashishg@stanford.edu
Interesting fact
Binary counter is combinatrorially equivalent to a de-multiplexer
If we could make large counters effiiciently, we would have a combinatorial template for a computer memory
Molecularly precise
Around 4X improvement over Silicon
Futuristic? Yes, but not as much as you might think
Ashish Goel,

22. Experiment by Barish et al
Made a counter (very similar to the one described above)
Unfortunately, lots of errors
Error-correction: huge challenge
Self-healing
Coding theory techniques vs bio-chemistry techniques
Ashish Goel,

23. The Source of Errors: Thermodynamics
GA = Activation energy
GB = Bond energy +
2GB GB GA GA
Correct Growth
Incorrect Growth
Rate of correct growth ¼ exp(-GA)
Probability of incorrect growth ¼ exp(-GA + GB)
Constraint: 2 GB > GA (system goes forward)
) Rate has quadratic dependence on error probability
) Time to reliably assemble an n £ n square ¼ n5
Ashish Goel,

24. Ashish Goel, ashishg@stanford.edu
Snake Tiles G2 G1 G3
G2a
G2b G4
G1b X2
G3b X1 X3
Each tile in the original system corresponds to four tiles in the new system
The internal glues are unique to this block
G1a
G3a
G4a
G4b
Ashish Goel,

25. Ashish Goel, ashishg@stanford.edu
How does this help?
First tile attaches with a weak binding strength
Ashish Goel,

26. Ashish Goel, ashishg@stanford.edu
How does this help?
First tile attaches with a weak binding strength
Second tile attaches and secures the first tile
Ashish Goel,

27. Ashish Goel, ashishg@stanford.edu
How does this help?
First tile attaches with a weak binding strength
Second tile attaches and secures the first tile
No Other tiles can attach without another nucleation error
Ashish Goel,

28. Ashish Goel, ashishg@stanford.edu
Preliminary Experimental Results
(Obtained by Chen, Goel, Schulman, Winfree)
Ashish Goel,

29. Ashish Goel, ashishg@stanford.edu

30. Ashish Goel, ashishg@stanford.edu

31. Ashish Goel, ashishg@stanford.edu

32. Ashish Goel, ashishg@stanford.edu

33. Four by Four Snake Tiles
Ashish Goel,

34. Four by Four Snake Tiles
Ashish Goel,

35. Four by Four Snake Tiles
Ashish Goel,

36. Four by Four Snake Tiles
Ashish Goel,

37. Four by Four Snake Tiles
Ashish Goel,

38. Four by Four Snake Tiles
Ashish Goel,

39. Four by Four Snake Tiles
Ashish Goel,

40. Four by Four Snake Tiles
Ashish Goel,

41. Four by Four Snake Tiles
Ashish Goel,

42. Four by Four Snake Tiles
Ashish Goel,

43. Four by Four Snake Tiles
Ashish Goel,

44. Four by Four Snake Tiles
Ashish Goel,

45. Ashish Goel, ashishg@stanford.edu
Analysis
Snake tile design extends to 2k£2k blocks.
Prevents tile propagation even after k+1 nucleation/growth errors
The error probability changes from p to roughly pk
We can assemble an N£N square in time O(N polylog N) and it remains stable for time W(N) (with high probability).
Resolution loss of O(log N)
Assuming tiles held by strength 3 do not fall off
Matches the time for ideal, irreversible assembly
Compare to N3 for basic proof-reading and N5 with no error-correction in the thermodynamic model [Chen, Goel; DNA ‘04]
Extensions, variations by Reif’s group, Winfree’s group, our group, and others
Recent result: Simple combinatorial criteria; Can avoid resolution loss by using third dimension [Chen, Goel, Luhrs; SODA ‘08]
Ashish Goel,

46. Ashish Goel, ashishg@stanford.edu
Open problem
Techniques for error reduction
Understand underlying bio-chemical primitives
Algorithms and combinatorial analysis
Probabilistic analysis
Serious simulations
Experimental verification
Challenges: What else could we do with self-assembly?
Antennas? Inductors? Origami followed by micro-scale assembly?
Ashish Goel,

47. Ashish Goel, ashishg@stanford.edu
Molecular machines
Ashish Goel,

48. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

49. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

50. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

51. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

52. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

53. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

54. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

55. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

56. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

57. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

58. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

59. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

60. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

61. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

62. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

63. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

64. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

65. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

66. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

67. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

68. Strand Invasion Strand Invasion (cont)
Ashish Goel,

69. Ashish Goel, ashishg@stanford.edu
Strand Invasion
Ashish Goel,

70. Ligation and restriction
Restriction enzymes: Recognize a specific DNA sequence and make a cut at a certain distance
Ligation: Attach two DNA sequences if they are held in place by another DNA strand
Ashish Goel,

71. Ashish Goel, ashishg@stanford.edu
DNA walker
[Turberfield et. Al. ’02]
Initial state: the walker (red strands) is attached
on top of the first strand.
Ashish Goel,

72. Ashish Goel, ashishg@stanford.edu
DNA walker
The first two strands attach together and get ligated.
Ashish Goel,

73. Ashish Goel, ashishg@stanford.edu
DNA walker
cut
An enzyme recognizes the binding site and
cut the two strands as illustrated.
Ashish Goel,

74. Ashish Goel, ashishg@stanford.edu
DNA walker
After the cutting, the walker moves to
the top of the second strand.
Ashish Goel,

75. Ashish Goel, ashishg@stanford.edu
Recent progress
Spiders that walk on a surface
Numerous other molecular machines in development
Our recent results: can embed Turing machine computation into this process
No experimental verification
Harbury lab -- potentially willing to host a student or a postdoc
Challenges
Modeling of bio-chemical primitives
New designs based on these primitives
Algorithmic and probabilistic analysis
Simulations
Experiments
Ashish Goel,

76. Ashish Goel, ashishg@stanford.edu
Outlook
Funding
EMT cluster in theoretical foundations within CISE
Expeditions in computing, etc.
Applications -- impressive in the short term; potentially revolutionary in the long term
Fabrication; Molecular machines
Computation in organic media; Genetic repair; drug delivery
Current status: like transistors in the 30’s
Intellectual:
Journal of Natural Computing
Dozens of exciting and important theory problems that I directly know of
Risk: there might be other technologies that are more useful
Solution: hedge our bets
Need larger intellectual community at Stanford?
Ashish Goel,

77. Ashish Goel, ashishg@stanford.edu
People
Winfree, Rothemund, Pierce at Caltech
Seeman at NYU
Reif at Duke
Turberfield at Cambridge
Stojanovich at Columbia
Yurke at Bell Labs
Second generation students
Many many more…
My role:
Small scale group; 1-2 students at any given time.
Around 25% of my research (may increase, depending on community)
Ashish Goel,

78. Example theoretical problems
Probabilistic
How does a counter remain stable?
How long does it take for a structure to form?
General techniques to analyze error rates?
Algorithmic
Can we simulate N restriction enzymes using O(1) enzymes?
Optimum tile system for assembling a shape?
Ashish Goel,