Skip to main content
SpringerLink
Log in
Book cover

Mathematical Control Theory I pp 73–93Cite as

Handling Biological Complexity Using Kron Reduction

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 461))

Abstract

We revisit a model reduction method for detailed-balanced chemical reaction networks based on Kron reduction on the graph of complexes. The resulting reduced model preserves a number of important properties of the original model, such as, the kinetics law and identity of the chemical species. For determining the set of chemical complexes for the deletion, we propose two alternative methods to the computation of error integral which requires numerical integration of the state equations. The first one is based on the spectral clustering method and the second one is based on the eigenvalue interlacing property of Kron reduction on the graph. The efficacy of the proposed methods is evaluated on two biological models.

This is a preview of subscription content, 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 EPUB and 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

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Cut-sets are the edges that connect the vertices of the different clusters.

  2. 2.

    For a detailed exposition on detailed-balanced CRNs with general kinetics, we refer interested readers to our work in [10].

  3. 3.

    For every \(i=1,\ldots , c\), the ith row of U corresponds to the ith vertex.

  4. 4.

    One can again perform the interative procedure as in Sect. 5.3 to obtain the best combination of complexes \(\mathcal {C}_\mathrm{red}\) from \(\overline{\mathcal C}_{red}\).

  5. 5.

    Here the complex composition matrix Z is given by an identity matrix.

References

  1. Biomodels database: an enhanced, curated and annotated resource for published quantitative kinetic models. http://www.ebi.ac.uk/biomodels-main/BIOMD0000000482

  2. B. Bollobas, Graduate Texts in Mathematics, Modern Graph Theory, vol. 184 (Springer, New York, 1998)

    Google Scholar 

  3. F. Döfler, F. Bullo, Kron reduction of graphs with applications to electrical networks. IEEE Trans. Circ. Syst. I 60(1), 150–163 (2013)

    Google Scholar 

  4. K. van Eunen, J.A.L. Kiewiet, H.V. Westerhoff, B.M. Bakker, Testing biochemistry revisited: how in vivo metabolism can be understood from in vitro enzyme kinetics. PLoS Comput. Biol. 8(4), e1002483 (2012)

    Article  Google Scholar 

  5. M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors -I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 43(10), 2229–2268 (1987)

    Article  Google Scholar 

  6. M. Feinberg, Complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49, 187–194 (1972)

    Article  MathSciNet  Google Scholar 

  7. W.H. Haemers, Interlacing eigenvalues and graphs. Linear Algebra Appl. 227–228, 593–616 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. F.J.M. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49, 172–186 (1972)

    Article  MathSciNet  Google Scholar 

  10. B. Jayawardhana, S. Rao, A.J. van der Schaft, Balanced Chemical Reaction Networks Governed by General Kinetics, in 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne, July 2012

    Google Scholar 

  11. T. Kanungo, D.M. Mount, N.S. Netanyahu, C.D. Piatko, R. Silverman, A.Y. Wu, An efficient k-means Clustering algorithm: analysis and implementation. IEEE Trans. Pattern Anal. Mach. Intell. 24(7), 881–892 (2002)

    Article  Google Scholar 

  12. G. Kron, Tensor Analysis of Networks (Wiley, 1939)

    Google Scholar 

  13. B.M. Maschke, A.J. van der Schaft, Port-controlled Hamiltonian Systems: Modelling Origins and System-Theoretic Properties, in Proceedings of the IFAC Symposium on NOLCOS, Bordeaux, pp. 282–288 (1992)

    Google Scholar 

  14. B.M. Maschke, A.J. van der Schaft, System-Theoretic Properties Of Port-controlled Hamiltonian Systems, Systems and Networks: Mathematical Theory and Applications, vol. II (Akademie-Verlag, Berlin, 1994), pp. 349–352

    Google Scholar 

  15. B.M. Maschke, A.J. van der Schaft, P.C. Breedveld, An intrinsic Hamiltonian formulation of network dynamics: nonstandard poisson structures and gyrators. J. Franklin Inst. 329, 923–966 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. B.M. Maschke, A.J. van der Schaft, P.C. Breedveld, An intrinsic Hamiltonian formulation of the dynamics of LC circuits. IEEE Trans. Circ. Syst. I 42, 73–82 (1995)

    Google Scholar 

  17. M. Meilă, W. Pentney, Clustering by Weighted Cuts in Directed Graphs, in Proceedings of 2007 SIAM International Conference on Data Mining (2007)

    Google Scholar 

  18. R. Noguchi et al., The selective control of glycolysis, gluconeogenesis and glycogenesis by temporal insulin patterns. Mol. Syst. Biol. 9, 664 (2013)

    Article  Google Scholar 

  19. J.F. Oster, A.S. Perelson, A. Katchalsky, Network dynamics: dynamic modeling of biophysical systems. Q. Rev. Biophys. 6(1), 1–134 (1973)

    Article  Google Scholar 

  20. J.F. Oster, A.S. Perelson, Chemical reaction dynamics, part I: geometrical structure. Arch. Ration. Mech. Anal. 55, 230–273 (1974)

    Article  MathSciNet  Google Scholar 

  21. O. Radulescu, A.N Gorban, A. Zinovyev, V. Noel, Reduction of dynamical biochemical reaction networks in computational biology. Front. Genet. 3, 00131 (2012)

    Google Scholar 

  22. S. Rao, A.J. van der Schaft, K. van Eunen, B.M. Bakker, B. Jayawardhana, Model reduction of biochemical reaction networks. BMC Syst. Biol. 8, 52 (2014)

    Article  Google Scholar 

  23. S. Rao, A.J. van der Schaft, B. Jayawardhana, A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks. J. Math. Chem 51(9), 2401–2422 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Shi, J. Malik, Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)

    Google Scholar 

  25. D. Siegel, D. MacLean, Global stability of complex balanced mechanisms. J. Math. Chem. 27, 89–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. A.J. van der Schaft, D. Jeltsema, Port-Hamiltonian systems theory: an introductory overview. Found. Trends Syst. Control 1(2/3), 173–378 (2014)

    Article  Google Scholar 

  27. A.J. van der Schaft, \(\text{ L }_2\)-Gain and Passivity Techniques in Nonlinear Control, 2nd revised and enlarged edn., (Springer, London, 2000)

    Google Scholar 

  28. A.J. van der Schaft, B.M. Maschke, A Port-Hamiltonian Formulation of Open Chemical Reaction Networks, Advances in the Theory of Control, Signals and Systems, Lecture Notes in Control and Information Sciences (Springer, New York, 2011), pp. 339–348

    Google Scholar 

  29. A.J. van der Schaft, B.M. Maschke, The Hamiltonian formulation of energy conserving physical systems with external ports. Arch. Elektron. \(\ddot{\text{ U }}\)bertragungstech 49, 362–371 (1995)

    Google Scholar 

  30. A. van der Schaft, S. Rao, B. Jayawardhana, On the mathematical structure of balanced chemical reaction networks governed by mass action kinetics. SIAM J. Appl. Math. 73(2), 953–973 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. A.J. van der Schaft, S. Rao, B. Jayawardhana, A Network Dynamics Approach to Chemical Reaction Networks. arXiv1502.02247, submitted for publication (2015)

  32. A.J. van der Schaft, Characterization and partial synthesis of the behavior of resistive circuits at their terminals. Syst. Control Lett. 59, 423–428 (2010)

    Article  MATH  Google Scholar 

  33. U. von Luxburg, A tutorial on spectral clustering. Stat. Comput. 17(4) (2007)

    Google Scholar 

Download references

Acknowledgments

The research is supported by NWO Centres for Systems Biology.

Author information

Authors and Affiliations

  1. Engineering and Technology Institute Groningen, University of Groningen, Nijenborgh 4, 9747AG, Groningen, The Netherlands

    Bayu Jayawardhana & Ward Sikkema

  2. Ghent University Global Campus, 119 Songdomunhwa-ro, Yeonsu-gu, Incheon, 406-840, South Korea

    Shodhan Rao

  3. Department of Pediatrics and Systems Biology Centre for Energy Metabolism and Ageing University of Groningen, University Medical Center Groningen, Groningen, The Netherlands

    Barbara M. Bakker

Authors
  1. Bayu Jayawardhana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shodhan Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ward Sikkema
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Barbara M. Bakker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bayu Jayawardhana .

Editor information

Editors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    M. Kanat Camlibel

  2. Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA

    A. Agung Julius

  3. Department of Electrical Engineering, Indian Institute of Technology, Chennai, Tamil Nadu, India

    Ramkrishna Pasumarthy

  4. Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Jayawardhana, B., Rao, S., Sikkema, W., Bakker, B.M. (2015). Handling Biological Complexity Using Kron Reduction. In: Camlibel, M., Julius, A., Pasumarthy, R., Scherpen, J. (eds) Mathematical Control Theory I. Lecture Notes in Control and Information Sciences, vol 461. Springer, Cham. https://doi.org/10.1007/978-3-319-20988-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20988-3_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20987-6

  • Online ISBN: 978-3-319-20988-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation