Speed Faults in Computation by Chemical Reaction Networks

  • Ho-Lin Chen
  • Rachel Cummings
  • David Doty
  • David Soloveichik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8784)

Abstract

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. Assuming a fixed molecular population size and bimolecular reactions, CRNs are formally equivalent to population protocols, a model of distributed computing introduced by Angluin, Aspnes, Diamadi, Fischer, and Peralta (PODC 2004). The challenge of fast computation by CRNs (or population protocols) is to ensure that there is never a bottleneck “slow” reaction that requires two molecules (agent states) to react (communicate), both of which are present in low (O(1)) counts. It is known that CRNs can be fast in expectation by avoiding slow reactions with high probability. However, states may be reachable (with low probability) from which the correct answer may only be computed by executing a slow reaction. We deem such an event a speed fault. We show that the problems decidable by CRNs guaranteed to avoid speed faults are precisely the detection problems: Boolean combinations of questions of the form “is a certain species present or not?”. This implies, for instance, that no speed fault free CRN could decide whether there are at least two molecules of a certain species, although a CRN could decide this in “fast” expected time – i.e. speed fault free CRNs “can’t count.”

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References

  1. 1.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distributed Computing 18, 235–253 (2006), Preliminary version appeared in PODC 2004Google Scholar
  2. 2.
    Angluin, D., Aspnes, J., Eisenstat, D.: Stably computable predicates are semilinear. In: PODC 2006: Proceedings of the Twenty-fifth Annual ACM Symposium on Principles of Distributed Computing, pp. 292–299. ACM Press, New York (2006)Google Scholar
  3. 3.
    Angluin, D., Aspnes, J., Eisenstat, D.: Fast computation by population protocols with a leader. Distributed Computing 21(3), 183–199 (2008); Preliminary version appeared in Dolev, S. (ed.) DISC 2006. LNCS, vol. 4167, pp. 61–75. Springer, Heidelberg (2006)Google Scholar
  4. 4.
    Cardelli, L.: Strand algebras for DNA computing. Natural Computing 10(1), 407–428 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cardelli, L., Csikász-Nagy, A.: The cell cycle switch computes approximate majority. Scientific Reports 2 (2012)Google Scholar
  6. 6.
    Cardoza, E., Lipton, R.J., Meyer, A.R.: Exponential space complete problems for Petri nets and commutative semigroups (preliminary report). In: STOC 1976: Proceedings of the 8th Annual ACM Symposium on Theory of Computing, pp. 50–54. ACM (1976)Google Scholar
  7. 7.
    Chen, H.-L., Doty, D., Soloveichik, D.: Deterministic function computation with chemical reaction networks. Natural Computing (2013); Preliminary version appeared in DNA 2012. LNCS, vol. 7433, pp. 25–42. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Chen, Y.-J., Dalchau, N., Srinivas, N., Phillips, A., Cardelli, L., Soloveichik, D., Seelig, G.: Programmable chemical controllers made from DNA. Nature Nanotechnology 8(10), 755–762 (2013)CrossRefGoogle Scholar
  9. 9.
    Condon, A., Hu, A., Maňuch, J., Thachuk, C.: Less haste, less waste: On recycling and its limits in strand displacement systems. Journal of the Royal Society Interface 2, 512–521 (2011); Preliminary version appeared in DNA 17 2011. LNCS, vol. 6937, pp. 84–99. Springer, Heidelberg (2011)Google Scholar
  10. 10.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. American Journal of Mathematics 35(4), 413–422 (1913)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Doty, D.: Timing in chemical reaction networks. In: SODA 2014: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 772–784 (January 2014)Google Scholar
  12. 12.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  13. 13.
    Karp, R.M., Miller, R.E.: Parallel program schemata. Journal of Computer and System Sciences 3(2), 147–195 (1969)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Petri, C.A.: Communication with automata. Technical report, DTIC Document (1966)Google Scholar
  15. 15.
    Soloveichik, D., Cook, M., Winfree, E., Bruck, J.: Computation with finite stochastic chemical reaction networks. Natural Computing 7(4), 615–633 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Soloveichik, D., Seelig, G., Winfree, E.: DNA as a universal substrate for chemical kinetics. Proceedings of the National Academy of Sciences 107(12), 5393 (2008); Preliminary version appeared in DNA Computing. LNCS, vol. 5347, pp. 57–69. Springer, Heidelberg (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ho-Lin Chen
    • 1
  • Rachel Cummings
    • 2
  • David Doty
    • 2
  • David Soloveichik
    • 3
  1. 1.National Taiwan UniversityTaipeiTaiwan
  2. 2.California Institute of TechnologyPasadenaUSA
  3. 3.University of California,San FranciscoSan FranciscoUSA

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