Abstract
This paper extends the stochastic analysis of self assembly in DNA-based computation. The new analysis models an error-correcting technique called pulsing which is analogous to checkpointing in computer operation. The model is couched in terms of the well-known tiling models of DNA-based computation and focuses on the calculation of computation times, in particular the times to self assemble rectangular structures. Explicit asymptotic results are found for small error rates q, and exploit the connection between these times and the classical Hammersley process. Specifically, it is found that the expected number of pulsing stages needed to complete the self assembly of an N ×N square lattice is asymptotically \(2N\sqrt{q}\) as N →∞ within a suitable scaling. Simulation studies are presented which yield performance under more general assumptions.
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References
Baryshnikov, Y., Coffman, E., Momčilović, P.: DNA-based computation times. In: Proc. of the Tenth International Meeting on DNA Computing, Milan, Italy (2004)
Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proc. ACM Symp. Th. Comput., pp. 740–748 (2001)
Rothemund, P., Winfree, E.: The program-size complexity of self-assembled squares. In: Proc. ACM Symp. Th. Comput., pp. 459–468 (2001)
Winfree, E.: Complexity of restricted and unrestricted models of molecular computation. In: Lipton, R., Baum, E. (eds.) DNA Based Computing, pp. 187–198. Am. Math. Soc., Providence, RI (1996)
Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Kempe, D., de Espanés, P.M., Rothemund, P.: Combinatorial optimization problems in self-assembly. In: Proc. ACM Symp. Th. Comput., Montreal, Canada, pp. 23–32 (2002)
Adleman, L., Cheng, Q., Goel, A., Huang, M.D., Wasserman, H.: Linear self-assemblies: Equilibria, entropy, and convergence rates. In: Elaydi, Ladas, Aulbach (eds.) New progress in difference equations. Taylor & Francis, Abington (2004)
Baryshnikov, Y., Coffman, E., Momčilović, P.: Incremental self-assembly in the fluid limit. In: Proc. 38th Ann. Conf. Inf. Sys. Sci., Princeton, NJ (2004)
Baryshnikov, Y., Coffman, E., Winkler, P.: Linear self-assembly and random disjoint edge selection. Technical Report 03-1000, Electrical Engineering Dept., Columbia University (2004)
Baryshnikov, Y., Coffman, E., Momčilović, P.: Phase transitions and control in self assembly. In: Proc. Foundations of Nanoscience: Self-Assembled Architectures and Devices, Snowbird, UT (2004)
Coffman, J.E.G., Flatto, L., Wright, P.E.: A stochastic checkpoint optimization problem. SIAM J. Comput. 22, 650–659 (1993)
Wang, H.: Dominoes and AEA case of the decision problem. In: Proc. of the Symposium in the Mathematical Theory of Automata. Polytechnic Press, Brooklyn (1963)
Berger, R.: The undecidability of the domino problem. In: Memoirs of the American Mathematical Society, vol. 66 (1966)
Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena, CA (1998)
Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985)
Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Chen, J., Reif, J.H. (eds.) DAN 2003. LNCS, vol. 2943, pp. 126–144. Springer, Heidelberg (2004)
Chen, H.L., Goel, A.: Error free self-assembly with error prone tiles. In: Proceedings of the Tenth International Meeting on DNA Based Computers, Milan, Italy (2004)
Reif, J.H., Sahu, S., Yin, P.: Compact error-resilient computational dna tiling assemblies. In: Proceedings of the Tenth International Meeting on DNA Based Computers. LNCS, pp. 293–307. Springer, New York (2004)
Chen, H.L., Cheng, Q., Goel, A., Huang, M.-D., de Espanes, P.M.: Invadable self-assembly: Combining robustness with efficiency. In: ACM-SIAM Symposium on Discrete Algorithms (2004)
Fujibayashi, K., Murata, S.: A method of error suppression for self-assembling DNA tiles. In: Proceedings of the Tenth International Meeting on DNA Based Computers. LNCS, pp. 284–293. Springer, New York (2004)
Aldous, D., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Th. Rel. Fields 103, 199–213 (1995)
Baryshnikov, Y., Coffman, E., Yimwadsana, T.: Analysis of self-correcting self-assembly growth models. Technical Report 03-1001, Electrical Engineering Dept., Columbia University (2005)
Mao, C., Sun, W., Seeman, N.C.: Designed Two-Dimensional DNA Holliday Junction Arrays Visualized by Atomic Force Microscopy. J. Am. Chem. Soc. 121, 5437–5443 (1999)
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Baryshnikov, Y., Coffman, E., Seeman, N., Yimwadsana, T. (2006). Self-correcting Self-assembly: Growth Models and the Hammersley Process. In: Carbone, A., Pierce, N.A. (eds) DNA Computing. DNA 2005. Lecture Notes in Computer Science, vol 3892. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11753681_1
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DOI: https://doi.org/10.1007/11753681_1
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