Natural Computing

, Volume 13, Issue 3, pp 287–296 | Cite as

An investigation into irreducible autocatalytic sets and power law distributed catalysis

  • Wim Hordijk
  • Leonard Hasenclever
  • Jie Gao
  • Dilyana Mincheva
  • Jotun Hein


RAF theory has been established as a useful and formal framework for studying the emergence and evolution of autocatalytic sets. Here, we present several new and additional results on RAF sets. In particular, we investigate in more detail the existence, expected sizes, and composition of the smallest possible, or irreducible, RAF sets. Furthermore, we study a more realistic variant of the well-known binary polymer model in which the catalysis events are assigned according to a power law distribution. Together, these results provide further insights into the existence and structure of autocatalytic sets in simple models of chemical reaction systems, with possible implications for theories on the origin of life.


Origin of life Autocatalytic sets Power-law distribution 



This paper resulted from a student project at the annual Oxford Summer School on Computational Biology (OSSCB13). We thank the Department of Statistics of the University of Oxford for funding. The complete project report, including additional and more detailed results, can be found online at:


  1. Ashkenasy G, Jegasia R, Yadav M, Ghadiri MR (2004) Design of a directed molecular network. PNAS 101(30):10,872–10,877CrossRefGoogle Scholar
  2. Contreras DA, Pereira U, Hernández V, Reynaert B, Letelier JC (2011) A loop conjecture for metabolic closure. In: Advances in Artificial Life, ECAL 2011: Proceedings of the Eleventh European Conference on the Synthesis and Simulation of Living Systems, MIT Press, pp 176–183Google Scholar
  3. Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: from biological nets to the internet and WWW. Oxford University Press, OxfordCrossRefGoogle Scholar
  4. Dyson FJ (1985) Origins of life. Cambridge University Press, CambridgeGoogle Scholar
  5. Eigen M, Schuster P (1979) The hypercycle. Springer, BerlinCrossRefGoogle Scholar
  6. Filisetti A, Graudenzi A, Serra R, Villani M, De Lucrezia D, Füchslin RM, Kauffman SA, Packard N, Poli I (2011) A stochastic model of the emergence of autocatalytic cycles. J Syst Chem 2:2CrossRefGoogle Scholar
  7. Gánti T (2003) The principles of life. Oxford University Press, OxfordCrossRefGoogle Scholar
  8. Hordijk W (2013) Autocatalytic sets: from the origin of life to the economy. BioScience 63(11):877–881CrossRefGoogle Scholar
  9. Hordijk W, Steel M (2004) Detecting autocatalytic, self-sustaining sets in chemical reaction systems. J Theor Biol 227(4):451–461CrossRefMathSciNetGoogle Scholar
  10. Hordijk W, Steel M (2012a) Autocatalytic sets extended: dynamics, inhibition, and a generalization. J Syst Chem 3:5CrossRefGoogle Scholar
  11. Hordijk W, Steel M (2012b) Predicting template-based catalysis rates in a simple catalytic reaction model. J Theor Biol 295:132–138CrossRefMathSciNetGoogle Scholar
  12. Hordijk W, Steel M (2013) A formal model of autocatalytic sets emerging in an RNA replicator system. J Syst Chem 4:3CrossRefGoogle Scholar
  13. Hordijk W, Hein J, Steel M (2010) Autocatalytic sets and the origin of life. Entropy 12(7):1733–1742CrossRefGoogle Scholar
  14. Hordijk W, Kauffman SA, Steel M (2011) Required levels of catalysis for emergence of autocatalytic sets in models of chemical reaction systems. Int J Mol Sci 12(5):3085–3101CrossRefGoogle Scholar
  15. Hordijk W, Steel M, Kauffman S (2012) The structure of autocatalytic sets: evolvability, enablement, and emergence. Acta Biotheor 60(4):379–392CrossRefGoogle Scholar
  16. Hordijk W, Steel M, Kauffman S (2013) Autocatalytic sets: the origin of life, evolution, and functional organization. In: Pontarotti P (ed) Evolutionary biology: exobiology and evolutionary mechanisms. Springer, BerlinGoogle Scholar
  17. Hordijk W, Wills PR, Steel M (2014) Autocatalytic sets and biological specificity. Bull Math Biol 76(1):201–224CrossRefzbMATHMathSciNetGoogle Scholar
  18. Kauffman SA (1971) Cellular homeostasis, epigenesis and replication in randomly aggregated macromolecular systems. J Cybern 1(1):71–96CrossRefMathSciNetGoogle Scholar
  19. Kauffman SA (1986) Autocatalytic sets of proteins. J Theor Biol 119:1–24CrossRefGoogle Scholar
  20. Kauffman SA (1993) The origins of order. Oxford University Press, OxfordGoogle Scholar
  21. Kayala MA, Azencott CA, Chen JH, Baldi P (2011) Learning to predict chemical reactions. J Chem Inf Model 51:2209–2222CrossRefGoogle Scholar
  22. Lincoln TA, Joyce GE (2009) Self-sustained replication of an RNA enzyme. Science 323:1229–1232CrossRefGoogle Scholar
  23. Maturana H, Varela F (1980) Autopoiesis and cognition: the realization of the living. Reidel, DordrechtCrossRefGoogle Scholar
  24. Mossel E, Steel M (2005) Random biochemical networks: the probability of self-sustaining autocatalysis. J Theor Biol 233(3):327–336CrossRefMathSciNetGoogle Scholar
  25. NetworkX Developers (2013) NetworkX. networkx.github.ioGoogle Scholar
  26. Newman MEJ (2010) Networks: an Introduction. Oxford University Press, OxfordCrossRefGoogle Scholar
  27. Sievers D, von Kiedrowski G (1994) Self-replication of complementary nucleotide-based oligomers. Nature 369:221–224CrossRefGoogle Scholar
  28. Smith J, Steel M, Hordijk W (2014) Autocatalytic sets in a partitioned biochemical network. J Syst Chem 5:2CrossRefGoogle Scholar
  29. Sousa FL, Hordijk W, Steel M, Martin W (2014) Autocatalytic sets in the metabolic network of E. coli. J Syst Chem (under review)Google Scholar
  30. Steel M (2000) The emergence of a self-catalysing structure in abstract origin-of-life models. Appl Math Lett 3:91–95CrossRefMathSciNetGoogle Scholar
  31. Steel M, Hordijk W, Smith J (2013) Minimal autocatalytic networks. J Theor Biol 332:96–107CrossRefMathSciNetGoogle Scholar
  32. Taran O, Thoennessen O, Achilles K, von Kiedrowski G (2010) Synthesis of information-carrying polymers of mixed sequences from double stranded short deoxynucleotides. J Syst Chem 1:9CrossRefGoogle Scholar
  33. Vaidya N, Manapat ML, Chen IA, Xulvi-Brunet R, Hayden EJ, Lehman N (2012) Spontaneous network formation among cooperative RNA replicators. Nature 491:72–77CrossRefGoogle Scholar
  34. Vasas V, Fernando C, Santos M, Kauffman S, Sathmáry E (2012) Evolution before genes. Biol Direct 7:1CrossRefGoogle Scholar
  35. Zipf GK (1932) Selected studies of the principle of relative frequency in language. Harvard University Press, CambridgeCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Wim Hordijk
    • 1
  • Leonard Hasenclever
    • 2
  • Jie Gao
    • 3
  • Dilyana Mincheva
    • 4
  • Jotun Hein
    • 4
  1. 1.SmartAnalytiX.comLausanneSwitzerland
  2. 2.University of CambridgeCambridgeUK
  3. 3.University of North Carolina at Chapel HillChapel HillUSA
  4. 4.University of OxfordOxfordUK

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