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Autocatalysis in Reaction Networks

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Abstract

The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of “critical siphon.” We explore the combinatorics of critical siphons, with a view toward the persistence conjecture. We introduce the notions of “drainable” and “self-replicable” (or autocatalytic) siphons. We show that: Every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and nonautocatalytic weakly reversible networks are persistent. Our results clarify that the difficulties in proving the persistence conjecture are essentially due to competition between drainable and self-replicable siphons.

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References

  • Adleman L, Gopalkrishnan M, Huang MD, Moisset P, Reishus D (2008) On the mathematics of the law of mass action, preprint, http://arXiv.org:0810.1108

  • Anderson DF (2008) Global asymptotic stability for a class of nonlinear chemical equations. SIAM J Appl Math 68(5):1464–1476

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson DF, Shiu A (2010) The dynamics of weakly reversible population processes near facets. SIAM J Appl Math 70(6):1840–1858

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson DF (2011) A proof of the global attractor conjecture in the single linkage class case. SIAM J Appl Math 71(4):1487–1508

    Article  MathSciNet  MATH  Google Scholar 

  • Angeli D, De Leenheer P, Sontag E (2007) A Petri net approach to persistence analysis in chemical reaction networks. In: Queinnec I, Tarbouriech S, Garcia G, Niculescu SI (eds) Biology and control theory: current challenges, lecture notes in control and information sciences, vol. 357, Springer, Berlin. doi:10.1007/978-3-540-71988-5_9, pp 181–216

  • Cardelli L (2009) Strand algebras for DNA computing. In: DNA and molecular programming, lecture notes in computer science 5877:12–24

  • Chen H-L, Doty D, Soloveichik D (2012) Deterministic function computation with chemical reaction networks. In: Preliminary extended abstract in proceedings of DNA computing and molecular programming, 18(7433):25–42

  • Craciun G, Pantea C, Nazarov F (2013) Persistence and permanence of mass-action and power-law dynamical systems. SIAM J Appl Math 73(1):305–329

    Article  MathSciNet  MATH  Google Scholar 

  • Del Vecchio D, Ninfa AJ, Sontag ED (2008) Modular cell biology: retroactivity and insulation. Mol Syst Biol 4:161. doi:10.1038/msb4100204

  • Donnell P, Banaji M (2012) Local and global stability of equilibria for a class of chemical reaction networks. SIAM J Appl Dyn Syst 12(2):899–920

    Article  MathSciNet  MATH  Google Scholar 

  • Dyson F (1982) A model for the origin of life. J Mol Evolut 18:344–350

    Article  Google Scholar 

  • Eigen M, Schuster P (1977) The hypercycle: a principle of natural self-organization. Part A: emergence of the hypercycle. Naturwissenschaften 64:541–565

    Article  Google Scholar 

  • Feinberg M (1989) Necessary and sufficient conditions for detailed balancing in mass-action systems of arbitrary complexity. Chem Eng Sci 44(9):1819–1827

    Article  Google Scholar 

  • Giri V, Jain S (2012) The origin of large molecules in primordial autocatalytic reaction networks. PLoS ONE 7(1). doi:10.1371/journal.pone.0029546

  • Gnacadja G (2011) Reachability, persistence, and constructive chemical reaction networks (part I): reachability approach to the persistence of chemical reaction networks. J Math Chem 49:2117–2136

    Article  MathSciNet  MATH  Google Scholar 

  • Gopalkrishnan M (2011) Catalysis in reaction networks. Bull Math Biol 73(12):2962–2982

    Article  MathSciNet  MATH  Google Scholar 

  • Gopalkrishnan M, Miller E, Shiu A (2013) A geometric approach to the global attractor conjecture. SIAM J Appl Dyn Syst 13(2):758–797

  • Gopalkrishnan M, Miller E, Shiu A (2013) A projection argument for differential inclusions, with application to mass-action kinetics. SIGMA 9:25

  • Guldberg CM, Waage P (1986) Studies concerning affinity. J Chem Educ 63:1044

    Article  Google Scholar 

  • Hordijk W, Steel M (2004) Detecting autocatalytic, self-sustaining sets in chemical reaction systems. J Theor Biol 227(4):451–461

    Article  MathSciNet  Google Scholar 

  • Hordijk W, Hein J, Steel M (2010) Autocatalytic sets and the origin of life. Entropy 12(7):1733–1742

    Article  Google Scholar 

  • Hordijk W, Kauffman S, Steel M (2011) Required levels of catalysis for emergence of autocatalytic sets in models of chemical reaction systems. Int J Mol Sci 12(5):3085–3101

    Article  Google Scholar 

  • Hordijk W, Steel M (2012) Predicting template-based catalysis rates in a simple catalytic reaction model. J Theor Biol 295:132–138

    Article  MathSciNet  Google Scholar 

  • Hordijk W, Steel M, Kauffman S (2012) The structure of autocatalytic sets: evolvability, enablement, and emergence. Acta Biotheoretica 60(4):379–392

    Article  Google Scholar 

  • Horn FJM (1974) The dynamics of open reaction systems. In: Mathematical aspects of chemical and biochemical problems and quantum chemistry (New York), proceedings of SIAM-AMS symposium. Appl. Math., vol. VIII

  • Jain S, Krishna S (1998) Autocatalytic sets and the growth of complexity in an evolutionary model. Phys Rev Lett 81:5684–5687

    Article  Google Scholar 

  • Jain S, Krishna S (2000) A model for the emergence of cooperation, interdependence, and structure in evolving networks. PNAS 98:543–547

    Article  Google Scholar 

  • Kauffman S (1995) At home in the universe: the search for the laws of self-organization and complexity. Oxford University Press, ISBN 0-19-509599-5

  • Kauffman S (1971) Cellular homeostasis, epigenesis and replication in randomly aggregated macromolecular systems. J Cybern 1:71–96

    Article  Google Scholar 

  • Kauffman KJ, Prakash P, Edwards JS (2003) Advances in flux balance analysis. Curr Opin Biotechnol 14(5):491–496

    Article  Google Scholar 

  • Lotka AJ (1925) Elements of physical biology. Williams and Wilkins

  • Mossel E, Steel M (2005) Random biochemical networks: the probability of self-sustaining autocatalysis. J Theor Biol 233(3):327–336

    Article  MathSciNet  Google Scholar 

  • Orth JD, Thiele I (2010) What is flux balance analysis? Nat Biotechnol 28:245–248

    Article  Google Scholar 

  • Pantea C (2012) On the persistence and global stability of mass-action systems. SIAM J Math Anal 44(3):1636–1673

    Article  MathSciNet  MATH  Google Scholar 

  • Petri C (1962) Kommunikation mit Automaten, Ph. D. Thesis. University of Bonn

  • Phillips A, Cardelli L (2009) A programming language for composable DNA circuits. J R Soc Interface 6(Suppl. 4):S419–S436

    Article  Google Scholar 

  • Qian L, Winfree E (2011) A simple DNA gate motif for synthesizing large-scale circuits. J R Soc Interface 8(62):1281–1297

    Article  Google Scholar 

  • Rabinovich Y, Sinclair A, Wigderson A (1992) Quadratic dynamical systems. In: Proceedings of the 33rd annual IEEE symposium on foudations of computer science, pp 304–313

  • Rozenberg G (1990) Advances in Petri nets. Springer, Berlin

    MATH  Google Scholar 

  • Savageau MA, Voit EO, Irvine DH (1987) Biochemical systems theory and metabolic control theory: 1. Fundamental similarities and differences. Math Biosci 86(2):127–145

    Article  MathSciNet  MATH  Google Scholar 

  • Schrijver A (1986) Theory of linear and integer programming. Wiley, London

    MATH  Google Scholar 

  • Shiu A, Sturmfels B (2010) Siphons in chemical reaction networks. Bull Math Biol 72(6):1448–1463

    Article  MathSciNet  MATH  Google Scholar 

  • Soloveichik D, Cook M, Winfree E, Bruck S (2008) Computation with finite stochastic chemical reaction networks. Nat Comput 7(4):615–633

    Article  MathSciNet  MATH  Google Scholar 

  • Soloveichik D, Seelig G, Winfree E (2010) DNA as a universal substrate for chemical kinetics. Proc Natl Acad Sci 107(12):5393–5398

    Article  Google Scholar 

  • Steel M (2000) The emergence of a self-catalysing structure in abstract origin-of-life models. Appl Math Lett 3:91–95

    Article  MathSciNet  MATH  Google Scholar 

  • Turberfield AJ, Yurke B, Winfree E (2007) Engineering entropy-driven reactions and networks catalyzed by DNA. Science 318(5853):1121–1125

    Article  Google Scholar 

  • Volterra V (1926) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem Acad Lincei Rom 2:31–113

    MATH  Google Scholar 

  • Yin P, Choi HMT, Calvert CR, Pierce NA (2008) Programming biomolecular self-assembly pathways. Nat Lett 451(7176):318–322

    Article  Google Scholar 

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Acknowledgments

We thank Jaikumar Radhakrishnan for pointing out that our proof of Theorem 4.9 showed a result mildly stronger (\(\mathbb {R}^n_{<0} \subseteq \mathrm{cone }(\mathrm{rows }(A))\)) than what we had asserted in a previous draft (\(\mathrm{cone }(\mathrm{rows }(A)) \cap \mathbb {R}^n_{<0}\) is nonempty). We thank the referees for useful comments.

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Authors and Affiliations

  1. Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Gachibowli, Hyderabad, India

    Abhishek Deshpande

  2. School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India

    Manoj Gopalkrishnan

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  1. Abhishek Deshpande
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  2. Manoj Gopalkrishnan
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Correspondence to Manoj Gopalkrishnan.

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Deshpande, A., Gopalkrishnan, M. Autocatalysis in Reaction Networks. Bull Math Biol 76, 2570–2595 (2014). https://doi.org/10.1007/s11538-014-0024-x

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  • DOI: https://doi.org/10.1007/s11538-014-0024-x

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