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Programmability of Chemical Reaction Networks

  • Matthew CookEmail author
  • David Soloveichik
  • Erik Winfree
  • Jehoshua Bruck
Chapter
Part of the Natural Computing Series book series (NCS)

Abstract

Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a well-stirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and Boolean Logic Circuits, Vector Addition Systems, Petri nets, Gate Implementability, Primitive Recursive Functions, Register Machines, Fractran, and Turing Machines. A theme to these investigations is the thin line between decidable and undecidable questions about SCRN behavior.

Keywords

Search Tree Turing Machine Boolean Circuit Clock Module Register Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361 CrossRefGoogle Scholar
  2. 2.
    Arkin AP, Ross J, McAdams HH (1998) Stochastic kinetic analysis of a developmental pathway bifurcation in phage-l Escherichia coli. Genetics 149:1633–1648 Google Scholar
  3. 3.
    Gibson M, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem A 104:1876–1889 CrossRefGoogle Scholar
  4. 4.
    Guptasarma P (1995) Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli? Bioessays 17:987–997 CrossRefGoogle Scholar
  5. 5.
    Levin B (1999) Genes VII. Oxford University Press, Oxford Google Scholar
  6. 6.
    McAdams HH, Arkin AP (1997) Stochastic mechanisms in gene expression. Proc Natl Acad Sci 94:814–819 CrossRefGoogle Scholar
  7. 7.
    Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 297:1183–1185 CrossRefGoogle Scholar
  8. 8.
    Suel GM, Garcia-Ojalvo J, Liberman LM, Elowitz MB (2006) An excitable gene regulatory circuit induces transient cellular differentiation. Nature 440:545–550 CrossRefGoogle Scholar
  9. 9.
    Esparza J, Nielsen M (1994) Decidability issues for Petri nets—a survey. J Inf Process Cybern 3:143–160 Google Scholar
  10. 10.
    Karp RM, Miller RE (1969) Parallel program schemata. J Comput Syst Sci 3(4):147–195 zbMATHMathSciNetGoogle Scholar
  11. 11.
    Conway JH (1972) Unpredictable iterations. In: Proceedings of the 1972 number theory conference. University of Colorado, Boulder, pp 49–52 Google Scholar
  12. 12.
    Conway JH (1987) Fractran: a simple universal programming language for arithmetic. Springer, New York, chap 2, pp 4–26 Google Scholar
  13. 13.
    Minsky M (1967) Computation: finite and infinite machines. Prentice–Hall, New Jersey zbMATHGoogle Scholar
  14. 14.
    Soloveichik D, Cook M, Winfree E, Bruck J (2008) Computation with finite stochastic chemical reaction networks. Nat Comput. doi: 10.1007/s11047-008-9067-y Google Scholar
  15. 15.
    Zavattaro G, Cardelli L (2008) Termination problems in chemical kinetics. In: van Breugel F, Chechik M (eds) CONCUR. Lecture notes in computer science, vol 5201. Springer, Berlin, pp 477–491 Google Scholar
  16. 16.
    Liekens AML, Fernando CT (2006) Turing complete catalytic particle computers. In: Proceedings of unconventional computing conference, York Google Scholar
  17. 17.
    Angluin D, Aspnes J, Eisenstat D (2006) Fast computation by population protocols with a leader. Technical Report YALEU/DCS/TR-1358, Yale University Department of Computer Science. Extended abstract to appear, DISC 2006 Google Scholar
  18. 18.
    Bennett CH (1982) The thermodynamics of computation—a review. Int J Theor Phys 21(12):905–939 CrossRefGoogle Scholar
  19. 19.
    Păun G (1995) On the power of the splicing operation. Int J Comput Math 59:27–35 zbMATHCrossRefGoogle Scholar
  20. 20.
    Kurtz SA, Mahaney SR, Royer JS, Simon J (1997) Biological computing. In: Hemaspaandra LA, Selman AL (eds) Complexity theory retrospective II. Springer, Berlin, pp 179–195 Google Scholar
  21. 21.
    Cardelli L, Zavattaro G (2008) On the computational power of biochemistry. In: Horimoto K, Regensburger G, Rosenkranz M, Yoshida H (eds) AB. Lecture notes in computer science, vol 5147. Springer, Berlin, pp 65–80 Google Scholar
  22. 22.
    Berry G, Boudol G (1990) The chemical abstract machine. In Proceedings of the 17th ACM SIGPLAN–SIGACT annual symposium on principles of programming languages, pp 81–94 Google Scholar
  23. 23.
    Păun G, Rozenberg G (2002) A guide to membrane computing. Theor Comput Sci 287:73–100 zbMATHCrossRefGoogle Scholar
  24. 24.
    Magnasco MO (1997) Chemical kinetics is Turing universal. Phys Rev Lett 78:1190–1193 CrossRefGoogle Scholar
  25. 25.
    Hjelmfelt A, Weinberger ED, Ross J (1991) Chemical implementation of neural networks and Turing machines. Proc Natl Acad Sci 88:10983–10987 zbMATHCrossRefGoogle Scholar
  26. 26.
    Petri CA (1962) Kommunikation mit Automaten. Technical Report Schriften des IMM No 2. Institut für Instrumentelle Mathematik, Bonn Google Scholar
  27. 27.
    Goss PJE, Peccoud J (1998) Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. Proc Natl Acad Sci USA 95:6750–6755 CrossRefGoogle Scholar
  28. 28.
    Mayr EW (1981) Persistence of vector replacement systems is decidable. Acta Inform 15:309–318 zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Sacerdote GS, Tenney RL (1977) The decidability of the reachability problem for vector addition systems (preliminary version). In: 9th annual symposium on theory of computing, Boulder, pp 61–76 Google Scholar
  30. 30.
    Post EL (1941) On the two-valued iterative systems of mathematical logic. Princeton University Press, New Jersey Google Scholar
  31. 31.
    Cook M (2005) Networks of relations. PhD thesis, California Institute of Technology Google Scholar
  32. 32.
    Rosier LE, Yen H-C (1986) A multiparameter analysis of the boundedness problem for vector addition systems. J Comput Syst Sci 32:105–135 zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Skolem T (1923) Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich. Videnskapsselskapets Skrifter. 1. Matematisk-naturvidenskabelig Klasse, 6 Google Scholar
  34. 34.
    Gödel K (1931) Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatschefte Math Phys 38:173–198 CrossRefGoogle Scholar
  35. 35.
    Turing A (1936–1937) On computable numbers, with and application to the Entscheidungsproblem. Proc Lond Math Soc 42(2):230–265 Google Scholar
  36. 36.
    Ackermann W (1928) Zum hilbertschen Aufbau der reellen Zahlen. Math Ann 99:118–133 zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Herbrand J (1931) Sur la non-contradiction de l’arithmétique. J Reine Angew Math 166:1–8 Google Scholar
  38. 38.
    Gödel K (1934) On undecidable propositions of formal mathematical systems. In: Davis M (ed) The undecidable. Springer, Berlin, pp 39–74. Lecture notes taken by Kleene and Rosser at Princeton Google Scholar
  39. 39.
    Church A (1936) An unsolvable problem of elementary number theory. Am J Math 58:345–363 CrossRefMathSciNetGoogle Scholar
  40. 40.
    Péter R (1951) Rekursive funktionen. Akadémiai Kiadó, Budapest zbMATHGoogle Scholar
  41. 41.
    Gale D (1974) A curious nim-type game. Am Math Mon 81:876–879 zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Minsky ML (1961) Recursive unsolvability of Post’s problem of ‘tag’ and other topics in theory of Turing machines. Ann Math 74:437–455 CrossRefMathSciNetGoogle Scholar
  43. 43.
    Neary T, Woods D (2005) A small fast universal Turing machine. Technical Report NUIM-CS-2005-TR-12, Dept. of Computer Science, NUI Maynooth Google Scholar
  44. 44.
    Soloveichik D (2008) Robust stochastic chemical reaction networks and bounded tau-leaping. arXiv:0803.1030v1
  45. 45.
    Cook M (2004) Universality in elementary cellular automata. Complex Syst 15:1–40 zbMATHGoogle Scholar
  46. 46.
    Cook M, Rothemund PWK (2004) Self-assembled circuit patterns. In: Winfree E (ed) DNA computing 9, vol 2943. Springer, Berlin, pp 91–107 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Matthew Cook
    • 1
    Email author
  • David Soloveichik
  • Erik Winfree
  • Jehoshua Bruck
  1. 1.Institute of NeuroinformaticsUZH, ETH ZürichZürichSwitzerland

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