Skip to main content
SpringerLink
Log in

Computing Realizations of Reaction Kinetic Networks with Given Properties

  • Conference paper
  • First Online:
Book cover Coping with Complexity: Model Reduction and Data Analysis

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 75))

  • 1531 Accesses

Abstract

The solution to the problem of finding the reaction kinetic realization of a given system obeying the mass action law containing the minimal/maximal number of reactions and complexes is shown in this paper. The proposed methods are based on Mixed Integer Linear Programming where the mass action kinetics is encoded into the linear constraints. Although the problems are NP-hard in the current setting, the developed algorithms give a usable answer to some of the questions first raised in [1].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hárs, V., Tóth, J.: On the inverse problem of reaction kinetics. In: Qualitative Theory of Differential Equations, Farkas, M., Hatvani L. (eds.). North-Holland, Amsterdam (1981) 363–379

    Google Scholar 

  2. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, NY (2000)

    MATH  Google Scholar 

  3. Samardzija, N., Greller, G.D., Wassermann, E.: Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems. Journal of Chemical Physics 90(4) (1989) 2296–2304

    Article  MathSciNet  Google Scholar 

  4. Horn, F., Jackson, R.: General mass action kinetics. Archive for Rational Mechanics and Analysis 47 (1972) 82–116

    Article  MathSciNet  Google Scholar 

  5. Haag, J., Wouver, A., Bogaerts, P.: Dynamic modeling of complex biological systems: a link between metabolic and macroscopic description. Mathematical Biosciencs 193 (2005) 25–49

    Article  MATH  Google Scholar 

  6. Sonta, E.: Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Autom. Control 46 (2001) 1028–1047

    Article  Google Scholar 

  7. Angeli. D.: A tutorial on chemical network dynamics. European Journal of Control 15 (2009) 398–406

    Google Scholar 

  8. Chellaboina, V., Bhat, S.P., Haddad, W.M., Bernstein., D.S.: Modeling and Analysis of Mass-Action Kinetics – Nonnegativity, Realizability, Reducibility, and Semistability. IEEE Control Systems Magazine 29 (2009) 60–78

    Google Scholar 

  9. Schnell, S., Chappell, M.J., Evans, N.D., Roussel, M.R.: The mechanism distinguishability problem in biochemical kinetics: The single-enzyme, single-substrate reaction as a case study. Comptes Rendus Biologies 329 (2006) 51–61

    Article  Google Scholar 

  10. Craciun, G., Pantea, C.: Identifiability of chemical reaction networks. Journal of Mathematical Chemistry 44 (2008) 244–259

    Article  MATH  MathSciNet  Google Scholar 

  11. Szederkényi, G.: Comment on Identifiability of chemical reaction networks by G. Craciun and C. Pantea. Journal of Mathematical Chemistry 45 (2009) 1172–1174

    Article  MATH  Google Scholar 

  12. Floudas, C.A.: Nonlinear and mixed-integer optimization. Oxford University Press, London (1995)

    MATH  Google Scholar 

  13. Gorban, A.N., Karlin I.V.: Method of invariant manifold for chemical kinetics. Chemical Engineering Science 58 (2003) 4751–4768

    Article  Google Scholar 

  14. Hardin, H.M., can Schuppen, J.H.: System reduction of nonlinear positive systems by linearization and truncation. Lecture Notes in Control and Information Sciences 341 (2006) 431–438

    Google Scholar 

  15. Szederkenyi, G.: Computing sparse and dense realizations of reaction kinetic systems. J. Math. Chem. 47(2) (2010) 551–568

    Article  MATH  MathSciNet  Google Scholar 

  16. Hangos, K.M., Szederkényi, G.: Special Positive Systems: the QP and the Reaction Kinetic System Class. In: Preprints of the Workshop on Systems and Control Theory in honor of József Bokor on his 60th Birthday, Hungarian Academy of Sciences (2008)

    Google Scholar 

  17. Feinberg, M.: Lectures on chemical reaction networks, Notes of lectures given at the Mathematics Research Centre, University of Wisconsin (1979)

    Google Scholar 

  18. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer, Berlin (2001)

    MATH  Google Scholar 

  19. Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors – I. The deficiency zero and deficiency one theorems. Chemical Engineering Science 42(10) (1987) 2229–2268

    Google Scholar 

  20. Gorban A.N., Karlin, I.V., Zinovyev, A.Y.: Invariant grids for reaction kinetics. Physica A 33 (2004) 106–154

    Article  Google Scholar 

  21. Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM Journal on Applied Mathematics 65(5) (2005) 1526–1546

    Article  MATH  MathSciNet  Google Scholar 

  22. Erdi, P., Tóth, J.: Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models. Manchester University Press, Manchester; Princeton University Press, Princeton (1989)

    Google Scholar 

  23. Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors – II. Multiple steady states for networks of deficiency one. Chemical Engineering Science 43 (1988) 1–25

    Google Scholar 

  24. Nemhauser, G.L., Wolsey, L. A.: Integer and Combinatorial Optimization. Wiley, NY (1988)

    MATH  Google Scholar 

  25. Löfberg, J.: YALMIP : A Toolbox for Modeling and Optimization in MATLAB In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)

    Google Scholar 

  26. Makhorin, A.: GNU Linear Programming Kit. Reference Manual. Version 4.10 (2006)

    Google Scholar 

  27. Homlstrm, K., Edvall, M.M., Gran, A.O.: TOMLAB for large-scale robust optimization. In: Nordic MATLAB Conference (2003)

    Google Scholar 

  28. Aykanat, C., Pinar, A.: Permuting sparse rectangular matrices into block-diagonal form. SIAM J. Scientific Computing 25 (2004) 1860–1879

    Article  MATH  MathSciNet  Google Scholar 

  29. Donoho, D.L., Tanner, J.: Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. of the National Academy of Sciences of the USA (PNAS) 102 (2005) 9446–9451

    Article  MathSciNet  Google Scholar 

  30. Donoho, D.L.: For most large undetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Communications on Pure and Applied Mathematics 59 (2006) 903–934

    Google Scholar 

  31. Lotka, A.J.: Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42 (1920) 1595–1599

    Article  Google Scholar 

  32. Wilhelm, T.: The smallest chemical reaction system with bistability. BMC Systems Biology 3 (2009) 90

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Hungarian National Research Fund (OTKA K-67625). The first author is a grantee of the Bolyai scholarship of the Hungarian Academy of Sciences.

Author information

Authors and Affiliations

  1. Process Control Research Group, Systems and Control Laboratory, Computer and Automation Research Institute, Hungarian Academy of Sciences, 1518, 63, Budapest, Hungary

    Gábor Szederkényi, Katalin M. Hangos & Dávid Csercsik

  2. Department of Electrical Engineering and Information Systems, University of Pannonia, 8200, Egyetem u. 10, Veszprém, Hungary

    Katalin M. Hangos

Authors
  1. Gábor Szederkényi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katalin M. Hangos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dávid Csercsik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Katalin M. Hangos .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  2. Dept. Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200a, Leuven, 3001, Belgium

    Dirk Roose

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Szederkényi, G., Hangos, K.M., Csercsik, D. (2011). Computing Realizations of Reaction Kinetic Networks with Given Properties. In: Gorban, A., Roose, D. (eds) Coping with Complexity: Model Reduction and Data Analysis. Lecture Notes in Computational Science and Engineering, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14941-2_13

Download citation

Publish with us

Policies and ethics

Search

Navigation