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Combining self-healing and proofreading in self-assembly

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Abstract

Molecular self-assembly is a promising approach to bottom-up fabrication of complex structures. A major impediment to the practical use of self-assembly to create complex structures is the high rate of error under existing experimental conditions. Recent theoretical work on algorithmic self-assembly has shown that under a realistic model of tile addition and detachment, error correcting tile sets are possible that can recover from the attachment of incorrect tiles during the assembly process. An orthogonal type of error correction was recently considered as well: whether damage to a completed structure can be repaired. It was shown that such self-healing tile sets are possible. However, these tile sets are not robust to the incorporation of incorrect tiles. It remained an open question whether it is possible to create tile sets that can simultaneously resist wholesale removal of tiles and the incorporation of incorrect ones. Here we present a method for converting a tile set producing a pattern on the quarter plane into a tile set that makes the same pattern (at a larger scale) but is able to withstand both of these types of errors.

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Notes

  1. Some backward growth can occur as a result of insufficient attachments in a growing complex that has not been subjected to wholesale tile removal. Chen and Goel’s construction handles this type of backward growth but not the more extensive incorrect backward growth possible after tile removal.

  2. Allowing self-assembly to start from a preexisting seed boundary as in this paper, rather than from a single seed tile as in (Winfree 2006), actually permits the use of a simpler transformation that produces a scale-up factor of just 2.

  3. This formulation ignores the initiation free energy of hybridization, which is non-negligible. See (Winfree 1998b) for details of how this free energy can be treated, yielding a model that is formally identical, but with slightly altered physical meanings for G mc and k f .

  4. Assuming f =  r 2, since r 1 ≫ f, if a tile is added that bonds only with strength 1, it falls off very quickly as it should to obey the aTAM. Tiles attached with strength 2 stick much longer, allowing an opportunity for other tiles to attach to them. Once a tile is bonded with total strength 3, it is very unlikely to dissociate (unless surrounding tiles fall off first).

  5. The approach here and in (Chen and Goel 2005) should be contrasted with the approach in (Winfree 1998b) where the effort is to maximize the “rate of growth” f − r 2 while maintaining a low error rate per tile. While for the tile systems considered there, the rate of growth is indeed proportional to f − r 2, the tile systems considered here and in (Chen and Goel 2005) can grow quickly even assuming f =  r 2. This is possible because of strong bonds that bias the assembly forward even if f =  r 2.

  6. We use the standard asymptotic notation defined as follows: f(x) = O(g(x)) means that that there is c > 0 such that f(x) ≤  c·g(x) for large enough x. Similarly, f(x) =  Ω(g(x)) means that there is c > 0 such that f(x) ≥  c·g(x) for large enough x. We write f(x) = Θ(g(x)) if f(x) = O(g(x)) and f(x) = Ω(g(x)).

  7. The Markov inequality states that for any non-negative random variable X, Pr[X ≥  a] ≤  E[X]/a where E[X] is the expected value of X.

  8. At the reflecting barrier the expected time to take a step is twice as large since only the forward direction is possible. However, this does not affect the asymptotic results.

  9. See (Feller 1968) for the general form of the expected duration of 1D discrete time random walks, from which the above expression is derived.

References

  • Adleman LM, Cheng Q, Goel A, Huang M-DA (2001) Running time and program size for self-assembled squares. In ACM Symposium on theory of computing (STOC), 740–748

  • Aggarwal G, Cheng Q, Goldwasser MH, Kao M-Y, de Espanés PM, Schweller RT (2005) Complexities for generalized models of self-assembly. SIAM J Comput 34:1493–1515

    Article  MATH  MathSciNet  Google Scholar 

  • Barish RD, Rothemund PWK, Winfree E (2005) Two computational primitives for algorithmic self-assembly: Copying and counting. Nano Lett 5:2586–2592

    Article  Google Scholar 

  • Chen HL, Goel A (2005) Error free self-assembly using error prone tiles. In: Ferretti C, Mauri G, Zandron C (eds) DNA Computing 10, LNCS vol 3384. Berlin, Springer-Verlag, pp 62–75

  • Cook M, Rothemund PWK, Winfree E (2004) Self-assembled circuit patterns. In: Chen J, Reif J (eds) DNA Computing 9, LNCS vol 2943. Berlin, Springer-Verlag, pp 91–107

  • Feller W (1968) An introduction to probability theory and its applications, vol 1. New York, Wiley

  • Rothemund PWK, Papakakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2:e424

    Article  Google Scholar 

  • Rothemund PWK, Winfree E (2000) The program-size complexity of self-assembled squares. In: ACM symposium on theory of computing (STOC), pp 459–468

  • LaBean TH, Yan H, Kopatsch J, Liu F, Winfree E, Reif JH, Seeman NC (2000) Construction, analysis, ligation, and self-assembly of DNA triple crossover complexes. J Am Chem Soc 122:1848–1860

    Article  Google Scholar 

  • Lagoudakis MG, LaBean TH (2000) 2-D DNA self-assembly for satisfiability. In: Winfree E, Gifford DK (eds) DNA Based Computers V, DIMACS vol 54. Providence, RI, American Mathematical Society, pp 141–154

  • Mao C, LaBean TH, Reif JH, Seeman NC (2000) Logical computation using algorithmic self-assembly of DNA triple-crossover molecules. Nature 407:493–496

    Article  Google Scholar 

  • Mao C, Sun W, Seeman NC (1999) Designed two-dimensional DNA holliday junction arrays visualized by atomic force microscopy. J Am Chem Soc 121:5437–5443

    Article  Google Scholar 

  • Reif J (1999) Local parallel biomolecular computing. In: Rubin H, Wood DH (eds) DNA Based Computers III, DIMACS vol 48. Providence, RI, American Mathematical Society, pp 217–254

  • Reif JH, Sahu S, Yin P (2005) Compact error-resilient computational DNA tiling assemblies. In: Ferretti C, Mauri G, Zandron C (eds) DNA Computing 10, LNCS vol 3384. Berlin, Springer-Verlag, pp 293–307

  • Schulman R, Winfree E (2005a) Programmable control of nucleation for algorithmic self-assembly. In: Ferretti C, Mauri G, Zandron C (eds) DNA Computing 10, LNCS vol 3384. Berlin, Springer-Verlag, pp 319–328. Extended abstract in DNA Computing 10; preprint of the full paper is cond-mat/0607317 on arXiv.org

  • Schulman R, Winfree E (2005b) Self-replication and evolution of DNA crystals. In: Capcarrere MS, Freitas AA, Bentley PJ, Johnson CG, Timmis J (eds) Advances in Artificial Life: 8th European Conference (ECAL), LNCS vol 3630. Berlin, Springer-Verlag, pp 734–743

  • Soloveichik D, Winfree E (2005) Complexity of compact proofreading for self-assembled patterns. In: DNA Computing 11. Berlin, Springer-Verlag

  • Soloveichik D, Winfree E (2007) Complexity of self-assembled shapes. SIAM J Comput 36:1544–1569

    Article  MATH  MathSciNet  Google Scholar 

  • Winfree E (1996) On the computational power of DNA annealing and ligation. In: Lipton RJ, Baum E B (eds) DNA Based Computers, DIMACS vol 27. Providence, RI, American Mathematical Society, pp 199–221

  • Winfree E (1998a) Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena

  • Winfree E (1998b) Simulations of computing by self-assembly. Technical Report CS-TR:1998.22, Caltech

  • Winfree E (2006) Self-healing tile sets. In: Chen J, Jonoska N, Rozenberg G (eds) Nanotechnology: science and computation. Springer-Verlag, Berlin, pp 55–78

    Chapter  Google Scholar 

  • Winfree E, Bekbolatov R (2004) Proofreading tile sets: error-correction for algorithmic self-assembly. In: Chen J, Reif J (eds) DNA Computing 9, LNCS vol 2943. Berlin, Springer-Verlag, pp 126–144

  • Winfree E, Liu F, Wenzler LA, Seeman NC (1998) Design and self-assembly of two dimensional DNA crystals. Nature 394:539–544

    Article  Google Scholar 

  • Winfree E, Yang X, Seeman NC (1998) Universal computation via self-assembly of DNA: some theory and experiments. In: Landweber LF, Baum EB (eds) DNA Based Computers II, DIMACS vol 44. Providence, RI, American Mathematical Society, pp 191–213

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Acknowledgments

We thank Ho-Lin Chen and Ashish Goel for insightful conversations and suggestions. This work was supported by NSF Grant No. 0523761.

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Authors and Affiliations

  1. Department of CNS, California Institute of Technology, Caltech MC 136-93, 1200 E. California Blvd, Pasadena, CA, 91126, USA

    David Soloveichik

  2. Institute of Neuroinformatics, Zurich, Switzerland

    Matthew Cook

  3. Department of CNS and CS, California Institute of Technology, Pasadena, CA, USA

    Erik Winfree

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  1. David Soloveichik
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  2. Matthew Cook
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Correspondence to David Soloveichik.

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Soloveichik, D., Cook, M. & Winfree, E. Combining self-healing and proofreading in self-assembly. Nat Comput 7, 203–218 (2008). https://doi.org/10.1007/s11047-007-9036-x

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