The Statistical Dynamics of Programmed Self-Assembly

1. Workshop on Swarming in Natural and Engineered Systems, Napa, August 2005 The Statistical Dynamics of Programmed Self-Assembly Eric Klavins Electrical Engineering University of Washington Collaborators: K. Böhringer, R. Ghrist Students: J. Bishop, S. Burden, J.M. McNew, N. Napp NSF: CAREER: Programmed Robotic Self-Assembly NSF(KB=PI): Modeling and Synthesis for 3D Self-Assembly

2. The Idea • How do local interactions give rise to global phenomena? • How do we program using only local interactions?

3. ? active tiles e.g. conformational switching Tile Self-Assembly Seems Like a Good Place to Start foam tiles, magnets, air hockey table. passive tiles • The system populates the minimum free energy states. • Can compute: Wang (1975) • Mesoscale (PDMS, Si, ...): Whitesides, (1999 etc.) and others • Can make from DNA: Winfree (1998)

4. rotating magnet assembly motor motor mount custom electronics fixed magnet IR transceiver And So Can Robots! Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005

5. 1cm Version 1.0 Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005 Initial test of latching mechanism Compare: Modular and Reconfigurable robots (e.g. Yim, Chirickjian, Murata, Rus, Lipson and many others). How do you program them?

6. We can automatically synthesize rules for any assembly problem. • Complexity characterized by partial order squashing. • Can reason about behaviors using temporal logic & model checking. • Examples: Assembly, ratcheting, self-replication, self-repair, distributed sensing,... [KGL,TAC] [K,FNANO] [K,ICMENS] [MK,InPrep] [BiK,InPrep] Graph Grammars [Klavins,Ghrist&Lipsy, TAC, To Appear] [30 years of gg research in CS] 1 1 2 1 3 A graph grammar with 60 rules for replicating strings [ICMENS2004]. 2 2 3

7. 4mers dimers the “right” 4mers hexagons “break rules” Example: Building Hexagons Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005

8. trimers first one at a time dimers first The Hexagon Grammar in Action 2x real speed.

9. A Well-MixedReaction Diffusion System The parts forget their initial velocities at approximately the same rate they “react” with each other. The velocity magnitudes are approximately Maxwellian.

10. number of number of etc. Macrostates for Self-Assembly ... an equivalence class of microstates. 4 2 1 1 multiplicity: entropy: free energy:

11. Statistical Dynamics The dynamics of the system are given by a Continuous Time Markov Process over the macrostates. basic rate multiplicity ... + + ... ... ... . . . “dog-flea model”

12. An Example: Sam’s Grammar

13. + 3 14 67 Obtaining Basic RatesNote: Obtaining from First Principles is Hard. multiplicity basic rate reaction . p(v,v’) For the price of a single set of simulations, you get an implicit open loop rate matrix for any number of parts.

14. a b c a a  b—c b a d a c  d—e b e a d j e i  j—k k a h h f f a g  h—i a i a a  f—g f g a Start with measured open loop rates: 1.1 Rates Induced by a Grammar 2.1 3.1 Goal structure 4.1 4.2 5.1 5.2 ... ... 6.4 ... ...

15. a b c a a  b—c b a d a c  d—e b e a d j e i  j—k k a h h f f a g  h—i a i a a  f—g f g a forward backward pkf kb c c a c’ a b’ + + recovery b b  a c’ (1-p)kf kb etc. f g’ g f’ a a + + g f Rates Induced by a Grammar Goal structure kf + kb

16. a b c a a  b—c b a d a c  d—e b e a d j e i  j—k k a h h f f a g  h—i a i a a  f—g backward forward f g a b b 0.5kf 0.5kb b d d a b d’ + + c e’ e e recovery c 0.5kf b a c 0.5kb d b b d  + a + b’  b b e’ c e e’ Rates Induced by a Grammar Goal structure kf + kb

17. a b c a a  b—c b a d a c  d—e b e a d j e i  j—k k a h h f f a g  h—i a i a a  f—g f g a g g b kf b d d +  f e f e etc. Rates Induced by a Grammar Goal structure kf + kb

18. a b c a a  b—c b a d a c  d—e b e a d j e i  j—k k a h h f f a g  h—i a i a a  f—g f g a Rates Induced by a Grammar Goal structure • More component types. • Changes and/or splits rates. • No longer admits a free-energy interpretation.

19. Example:  = {a a ! b-b} + + ... Note: Can use rates for an ODE model based on concentrations, but we loose important information this way.

20. An Approach toFinding the Best Grammar Klavins and McNew, In Preparation Problem: Optimize the probability that the system behaves like the specification F. (G0,,) Find the best initial graph (e.g. change the base pair sequence in a DNA strand). Change the rules (e.g. reprogram the robots). Tweak the rates (e.g. optimize binding efficiencies). p(F|) is continuous in . p(F|G0) and p(F|) are discrete.

21. F: (k) Optimizing  Example: Find k so that (k) behaves like the specification F half of the time. • Approach: • Sample behaviors of (k) to estimate |p(F|(k))-0.5| for a given k. • Use these samples to estimate the gradient and do gradient descent. A neighborhood automaton. Klavins and McNew, In preparation.

22. noise due to small sample size Results of Optimization • Only “finitely generated” properties can be checked. • Need a good model and an efficient simulation. • Can put  = (k1) [(k2) to combine/compare grammars. • We are building the software infrastructure needed to do this for large problems.

23. Other Systems 1 2 DNA Hybridization Reactions (UW RRF Grant) Cooperative Control (With R. Murray) Continuous distributed control coupled with diffusive exploration and graph grammars. Can tune binding strengths of strand segments. 3 MEMs 3D Self-Assembly (NSF Grant with K. Böhringer) Change surface properties to tune equilibrium distribution.

24. Conclusions We program our robots with graph grammars which are very cool. But grammars model possibility not probability. We can measure the “natural” rates of our systems. A grammar changes the natural dynamics of the system in a predictable way. We can specify the desired behavior of our systems in pCTL. We are beginning to understand how to optimize the probability that the systems behaves correctly.