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On the Complexity of Reconstructing Chemical Reaction Networks

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Abstract

The analysis of the structure of chemical reaction networks is crucial for a better understanding of chemical processes. Such networks are well described as hypergraphs. However, due to the available methods, analyses regarding network properties are typically made on standard graphs derived from the full hypergraph description, e.g. on the so-called species and reaction graphs. However, a reconstruction of the underlying hypergraph from these graphs is not necessarily unique. In this paper, we address the problem of reconstructing a hypergraph from its species and reaction graph and show NP-completeness of the problem in its Boolean formulation. Furthermore we study the problem empirically on random and real world instances in order to investigate its computational limits in practice.

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References

  1. Limboole. http://fmv.jku.at/limboole/index.html. Institute for Formal Models and Verification (2013)

  2. Alon, N., Galil, Z., Margalit, O., Naor, M.: Witnesses for boolean matrix multiplication and for shortest paths. In: Foundations of Computer Science, 1992. Proceedings of 33rd Annual Symposium, pp. 417–426. IEEE (1992)

  3. Andersen, J., Flamm, C., Merkle, D., Stadler P.: Maximizing output and recognizing autocatalysis in chemical reaction networks is NP-complete. J. Syst. Chem. 3(1) (2012)

  4. Barrett, C., Stump, A., Tinelli, C.: The SMT-LIB Standard: Version 2.0. In: Gupta, A., Kroening, D. (eds.) Proceedings of the 8th International Workshop on Satisfiability Modulo Theories (Edinburgh, UK) (2010)

  5. Benkö G., Centler F., Dittrich P., Flamm C., Stadler B.M.R., Stadler P.F.: A topological approach to chemical organizations. Alife 15, 71–88 (2009)

    Google Scholar 

  6. Biere, A.: Lingeling, plingeling, picosat and precosat at sat race 2010. FMV Report Series Technical Report 10(1) (2010)

  7. Biere, A., Heule, M., van Maaren, H., Walsh, T.: Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185 (2009)

  8. Borenstein, E., Kupiec, M., Feldman, M.W., Ruppin, E.: Large-scale reconstruction and phylogenetic analysis of metabolic environments. Proc. Natl. Acad. Sci. USA 105, 14482–14487 (2008)

    Google Scholar 

  9. Capobianco M., Molluzzo J.C.: Examples and counterexamples in graph theory. J. Graph Theory 2(3), 274–274 (1978)

    Article  MathSciNet  Google Scholar 

  10. Cottret, L., Wildridge, D., Vinson, F., Barrett, M.P., Charles, H., Sagot, M.-F., Jourdan, F.: Metexplore: a web server to link metabolomic experiments and genome-scale metabolic networks. Nucleic Acids Res. 38(suppl 2), W132–W137 (2010)

    Google Scholar 

  11. Craciun G., Feinberg M.: Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J. Appl. Math. 66, 1321–1338 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Moura L., Bjørner N.: Z3: An efficient SMT solver. Tools Algorithms Constr. Anal. Syst. (TACAS’) 08(4963), 337–340 (2008)

    Article  Google Scholar 

  13. Domijan M., Kirkilionis M.: Graph theory and qualitative analysis of reaction networks. Netw. Heterog. Media 3, 295–322 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eén, N., Sörensson, N.: An extensible sat-solver. In: Theory and Applications of Satisfiability Testing, pp. 333–336. Springer, New York (2004)

  15. Erdős P., Rényi A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl 5, 17–61 (1960)

    Google Scholar 

  16. Feinberg M.: Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49, 187–194 (1972)

    Article  MathSciNet  Google Scholar 

  17. Gallo G., Longo G., Pallottino S., Nguyen S.: Directed hypergraphs and applications. Discrete Appl. Math. 42(2), 177–201 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness (1979)

  19. Han J.D.J., Bertin N., Hao T., Goldberg D.S., Berriz G.F., Zhang L.V., Dupuy D., Walhout A.J., Cusick M.E., Roth F.P. et al.: Evidence for dynamically organized modularity in the yeast protein–protein interaction network. Nature 430(6995), 88–93 (2004)

    Article  Google Scholar 

  20. Horn F.J.M., Jackson R.: General mass action kinetics. Arch. Ration. Mech. Anal. 47, 81–116 (1972)

    Article  MathSciNet  Google Scholar 

  21. Ivanova A.N.: Conditions for uniqueness of stationary state of kinetic systems, related, to structural scheme of reactions. Kinet. Katal 20, 1019–1023 (1979)

    Google Scholar 

  22. Jeong H., Tombor B., Albert R., Oltvai Z., Barabási A.: The large-scale organization of metabolic networks. Nature 407(6804), 651–654 (2000)

    Article  Google Scholar 

  23. Kaltenbach, H.-M.: A unified view on bipartite species-reaction and interaction graphs for chemical reaction networks. Technical Report 1210.0320, arXiv (2012)

  24. Klamt S, Haus U-U, Theis F (2009) Hypergraphs and cellular networks. PLoS Comput Biol 5:e1000385

  25. Lacroix V., Cottret L., Thébault P., Sagot M.: An introduction to metabolic networks and their structural analysis. IEEE/ACM Trans. Comput. Biol. Bioinforma. 5(4), 594–617 (2008)

    Article  Google Scholar 

  26. McNaught, A., Wilkinson, A.: IUPAC. Compendium of Chemical Terminology, vol. 2. Blackwell Scientific Publications, Oxford (1997)

  27. Miettinen P., Mielikainen T., Gionis A., Das G., Mannila H.: The discrete basis problem. IEEE Trans. Knowl. Data Eng. 20(10), 1348–1362 (2008)

    Article  Google Scholar 

  28. Mincheva M., Roussel M.R.: A graph-theoretic method for detecting potential Turing bifurcations. J. Chem. Phys 125, 204102 (2006)

    Article  Google Scholar 

  29. Mincheva M., Roussel M.R.: Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models. J. Math. Biol. 55, 61–68 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shinar G., Feinberg M.: Concordant chemical reaction networks and the species-reaction graph. Math. Biosci. 241, 1–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Skiena, S.: Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (1990)

  32. Sörensson, N., Eén, N.: Minisat v1. 13-a sat solver with conflict-clause minimization. SAT 2005, 53 (2005)

  33. Soulé C.: Graphic requirements for multistationarity. ComplexUs 1, 123–133 (2003)

    Article  Google Scholar 

  34. Tanaka R.: Scale-rich metabolic networks. Phys. Rev. Lett. 94(16), 168101 (2005)

    Article  Google Scholar 

  35. Thomas, R.: On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. In: Della Dora, J., Demongeot, J., Lacolle, B. (eds.) Numerical Methods in the Study of Critical Phenomena, vol. 9, pp. 180–193. Springer, Berlin (1981)

  36. Tseitin, G.: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Structures in Constructive Mathematics and Mathematical Logic, Part II, Seminars in Mathematics (translated from Russian), Steklov Mathematical Institute, pp. 115–125 (1968)

  37. Tseitin G.: On the complexity of derivation in propositional calculus. Autom. Reason. Class. Papers Comput. Logic 2, 466–483 (1983)

    Google Scholar 

  38. Volpert, A., Ivanova, A.N.: Mathematical models in chemical kinetics. In: Mathematical Modeling, pp. 57–102. Nauka, Moscow (1987) (Russian)

  39. Wagner A., Fell D.A.: The small world inside large metabolic networks. Proc. R. Soc. Lond. B 268, 1803–1810 (2001)

    Article  Google Scholar 

  40. Watts D.J., Strogatz S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 409–410 (1998)

    Article  Google Scholar 

  41. Zeigarnik, A.V.: On hypercycles and hypercircuits in hypergraphs. In: Hansen, P., Fowler, P.W., Zheng, M. (eds.) Discrete Mathematical Chemistry, volume 51 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 377–383. American Mathematical Society, Providence (2000)

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Rolf Fagerberg, Daniel Merkle & Philipp Peters

  2. Institute for Theoretical Chemistry, University of Vienna, Währingerstraße 17, 1090, Vienna, Austria

    Christoph Flamm & Peter F. Stadler

  3. Bioinformatics Group, Department of Computer Science, Interdisciplinary Center for Bioinformatics, Härtelstraße 16-18, 04107, Leipzig, Germany

    Peter F. Stadler

  4. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Peter F. Stadler

  5. Fraunhofer Institute for Cell Therapy and Immunology, Perlickstraße 1, 04103, Leipzig, Germany

    Peter F. Stadler

  6. Center for non-coding RNA in Technology and Health, University of Copenhagen, Grønnegårdsvej 3, 1870, Frederiksberg C, Denmark

    Peter F. Stadler

  7. Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM, 87501, USA

    Peter F. Stadler

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  1. Rolf Fagerberg
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  2. Christoph Flamm
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  3. Daniel Merkle
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  4. Philipp Peters
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  5. Peter F. Stadler
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Correspondence to Daniel Merkle.

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Fagerberg, R., Flamm, C., Merkle, D. et al. On the Complexity of Reconstructing Chemical Reaction Networks. Math.Comput.Sci. 7, 275–292 (2013). https://doi.org/10.1007/s11786-013-0160-y

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  • DOI: https://doi.org/10.1007/s11786-013-0160-y

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