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Robust Self-Assembly of DNA

Robust Self-Assembly of DNA . Eduardo Abeliuk Dept. of Electrical Engineering Stanford University November 30, 2006. Agenda for today. Robust Self-Assembly: definitions and motivation Basic assembly model and examples “ Complexity of Self-Assembled Shapes ”

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Robust Self-Assembly of DNA

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  1. Robust Self-Assembly of DNA Eduardo Abeliuk Dept. of Electrical Engineering Stanford University November 30, 2006

  2. Agenda for today • Robust Self-Assembly: definitions and motivation • Basic assembly model and examples • “Complexity of Self-Assembled Shapes” D. Soloveichik, E.Winfree. DNA Computers 10, LNCS v.3384, 2005 • “Error Free Self-Assembly using Error Prone Tiles”, H. Chen, A. Goel. 10th Int. Meeting on DNA Based Computers, 2004. • “Self-Healing Tile Sets” E. Winfree, Nanotechnolgy: Science and computation, p.55-78, 2006

  3. Self-Assembly Theory • Self-assembly: no precise general definition • But roughly speaking: “ process by which an organized structure can spontaneously form from simpler parts” • Programming • Complexity • Fault-tolerance • Self-healing • Self-reproduction and evolution Schulman R., Winfree E., “Self-replication and evolution of DNA crystals” 2005.

  4. Self-assembly • Already present in nature • Inside cells • Robust self-assembly of organisms over 18 orders of magnitude in volume! • Bottom-up fabrication of complex structures: • Arbitrary shapes can be self-assembled (2D) • Enabled by DNA nanotechnology Rothemund PWK, “Folding DNA to create nanoscale shapes and patterns”, Nature 2006

  5. Rothemund PWK, “Folding DNA to create nanoscale shapes and patterns”, Nature 2006

  6. Rothemund PWK, “Folding DNA to create nanoscale shapes and patterns”, Nature 2006

  7. output: input: 01001101011 output: 01001101011 input: Another Motivation • Compute “along the way” • The self-assembly of a crystal can resemble a program that leaves the traces of its operations embedded in it. • The assembly of a 2D crystal can simulate a universal Turing machine!

  8. Robust self-assemblyof DNA • Do we need robustness? "In theory, there is no difference between theory and practice. But, in practice, there is." -Jan L.A. van de Snepscheut • Computing with DNA, and not transistors??

  9. The tile assembly model • Infinite lattice: • Z x Z Every position in the grid has a relative position associated: N(i,j)=(i,j+1) S(i,j)=(i,j-1) E(i,j)=(i+1,j) W(i,j)=(i-1,j) (i,j) N W E S

  10. A B C D B A B D D A C D Bond types and Tile Types • Our fundamental unit is a square tile with labellededges, or bond types. • We consider a set of bond types . (e.g., ={A,B,C,D,null}) • A reflection or rotation gives a different tile. So a tile type is a quadruple: and we have unlimited supply of them • Tiles types with identical edges can pair with each other. • We will represent tile types with different colors. All tile types for the set T. A B C D B A B D D A C D B A B D A B C D

  11. Tiles • A tile is a pair , i.e., it corresponds to a tile with certain tile type located in a certain position in our grid • A configuration is a set of tiles, such that there is exactly one tile in every location Configuration 2 Configuration 1

  12. Interaction between tiles • A strength function defines the interactions between two tiles. • We say a tile t1 interacts with its neighbor t2 with strength • Usually, only diagonal strength functions are considered, and the range of g is {0,1,2} g A B C D null A 1 0 0 0 0 B 0 1 0 0 0 C 0 0 2 0 0 D 0 0 0 1 0 null 0 0 0 0 0 B A B C A B C D A C C D

  13. The tile assembly model (aTAM) • A tile system is a quadruple i.e., it consist of • a set of tile types • a seed tile • a strength function • a binding threshold or “temperature” • Self-assembly is defined as a relation between configurations: A B

  14. Example of Tile Systems Sierpinski tile set • 7 types of tiles: 1 seed, 2 boundary (input) tiles, 4 rule tiles Boundary tiles Seed “Rule” tiles

  15. Sierpinski tile set Tiles

  16. Sierpinski tile set

  17. Binary counter • Tyle types:

  18. 8 7 6 5 4 3 2 1 Binary counter • Begin with seed • Continue with boundary tiles • Then “rule” tiles • Count upwards (binary) = 1 = 0 Tiles

  19. Square self-assembly • Example for 9x9 square • 41 Tile types

  20. From binary counter output terminal input More on assemblies • Tile additions are non-deterministic • Several locations for adding tiles • Several possible tiles could be added in one spot • Defininitions: • input sides • propagation (output) sides • terminal sides.

  21. Final Assembly Theorem • Definition: an assembly is locally deterministic if: • every tile addition has strength 2. • if tile at (i,j) and all tiles touching its propagation sides are removed, then there is only one tile type that can be added at (i,j) • Theorem: “If a tile set has one locally determinist assembly sequence, then the same final assembly is produced regardless of order of tile additions”.

  22. From theory to practice (biology) • Tiles are “do-able” in practice • DNA Nano-technology Winfree, E. et at. Design and self-assembly of two dimensional. DNA crystals. 1998.

  23. More on the technology • More tiles from DNA Hao Yan et al. “4x4 DNA Tile and Lattices: Characterization, Self-Assembly and Metallization of a Novel DNA Nanostructure Motif” 2003.

  24. Road ahead… • First paper (complexity): • Ties (Kolmogorov) computation of a shape with complexity of tile system that self-assembles it • Note: the former has nothing to do with self-assembly Robust self-assembly • Second paper (fault-tolerance): • how to avoid nucleation and growth errors • Third paper (self-healing): • how to avoid gross damage

  25. First paper • D. Soloveichik, E.Winfree, “Complexity of Self-Assembled Shapes”

  26. Coordinated shapes • Let S be a finite set of locations in Z2. S is a coordinated shape if it’s connected Coordinated shape This is not

  27. Transforming coordinated shapes • Scalings: • Translations:

  28. Shapes • Scale and translation equivalence relations on coordinated shapes define class of shapes coordinated shape 1 coordinated shape 3 coordinated shape 2 All belong to the same class of shapes =

  29. Computer Science Concepts • Kolmogorov complexity: • = size of the smallest program outputting the coordinated shape as a list of locations • Similarly

  30. Tile Complexity • The tile-complexity of a coordinated shape S is: n s.t. exists a tile system T of n tile types Ksa(S)= min that uniquely produces assembly A and S is the coordinated shape of A. n s.t. exists a tile system T of n tile types Ksa( )= min that uniquely produces assembly A and is the shape of A.

  31. Main Theorem • There exist constants such that for any shape , • To show the right inequality, the paper explicitly shows how to find an optimum tile system that assembles a given shape! (proof in paper) • The minimun number of bits required to store n tile types is i.e., same complexity as Kolmogorov complexity of shape!

  32. Corollary • (tile complexity) of shapes is uncomputable. i.e., given a shape, the minimun number of tiles required to assemble it cannot be computed • Formally, the following language is undecidable:

  33. Second paper • H. Chen, A. Goel, “Error Free Self-Assembly using Error Prone Tiles”

  34. Kinetic Tile Assembly Model (kTAM) • Stochastic model. • Add and remove tiles • Kinetics: • Two parameter:

  35. Snake proof reading • A simple one dimensional example will be used to illustrate the algorithm. • We will consider four tile types, with two bond types + null bond:

  36. Snake proof reading (2) • The input will consist of a structure of n+2 tiles • Our 1-D crystal will output the parity of the input. • input=“1111” (n=4) 0 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1

  37. Definition • Insufficient attachment (at ) is: A process where a tile attaches with strength one but before it falls, another tile attaches next to it (and now both are held by strength ). • They can cause two type of errors: • growth errors • nucleation errors.

  38. Growth Errors • A growth error is an invalid tile attachment to a “valid” position 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 Incorrect pairing

  39. Nucleation Errors • When a tile attaches to an incorrect position (small binding strength) 0 0 1 1 1 0 1 0 1 1 1 1 1 1 Correct pairing, but weak bonding..that then stabilizes

  40. Winfree-Bekbolatov proofreading system • Replace each tile with 2x2 blocks. • The internal glues are all unique to the 2x2 block • Corrects for growth but not nucleation errors.

  41. Snake proofreading • Replace each tile with 2x2 blocks. • The internal glues are all unique to the 2x2 block • Corrects for growth and nucleation errors.

  42. Robust parity check • Replace tiles with 2x2 blocks • Note location of strong bonds and null bonds 0T 0B X3 X2 X3 1T X4 1T X7 X6 X7 0T X8 0B X3 X2 X3 1T X4 1T X7 X6 X7 0T X8 0B X2 0B 1L X4 1B 1R X6 1B 1L X8 X5 0B 1R X2 0B 1L X4 1B 1R X6 1B 1L X8 0B 1R 1L 1R 1L 1R 1L 1R 1L 1R

  43. Insufficient Attachment Insufficient attachment No error Weak tile attaches Insufficient attachments 0T 0B X3 X2 X3 1T X4 0B X2 0B 1L X4 1B 1R X8 X5 0B 1R X2 0B 1L 1L 1R 1L 1R 1L 1R 1L 1R Continuous Markov Chain Model

  44. Insufficient attachment No error Insufficient attachments • 1 insufficient attachment is very unlikely, but over the course of n attachments, the probability of getting at least one insufficient attachment might become significant. • Snake-proofreading requires two insufficient attachments in close proximity to have an error than can propagate.

  45. 1 Insufficient attachment 2 Insufficient attachments No error Nucleation error improvement Cannot propagate with tau=2 unless another insufficient attachment occurs 0T 0B X3 X2 X3 1T X4 0B X2 0B 1L X4 1B 1R X8 X5 0B 1R X2 0B 1L 1L 1R 1L 1R 1L 1R 1L 1R

  46. General Snake proofreading • Previous example only considered one directional growth. • General method extends to: • L-bounded systems (growth S  N and E  W) • Replaces a tile by k x k block • Set of rules to construct internal bonds • All internal bonds are unique to the tile block • Most of them have strength 1, some have strength 0, some have strength 2.

  47. Example • Notice how tiles are constructed following a snake pattern. T4,1 T4,2 T4,3 T4,4 T3,1 T3,2 T3,3 T3,4 T2,1 T2,2 T2,3 T2,4 T1,1 T1,2 T1,3 T1,4

  48. Main analytical results • Theorem 1 With a 2k x 2k snaked tile system (for k sufficiently large) assuming we can set to be , an N x N square of blocks can be assembled in time and w.h.p no block errors happen for time after that. • Theorem 2 With a 2k x 2k snaked tile system ( ) assuming that we can set to be , an N x N square of blocks can be assembled in time and w.h.p no block errors happen for time after that. Informally, snaked proofreading results in tile systems which assemble quickly and remain stable for long time

  49. Simulation results (1) • Three systems simulated using xgrow • no proofreading, WB proofreading, snaked • 4x4 tile blocks

  50. Simulation results (2) • Simulations relaxed idealized modeling conditions • They corroborate analytical results

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