Skip to main content

Hybrid Reductions of Computational Models of Ion Channels Coupled to Cellular Biochemistry

  • Conference paper
  • First Online:
Computational Methods in Systems Biology (CMSB 2016)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 9859))

Included in the following conference series:

Abstract

Computational models of cellular physiology are often too complex to be analyzed with currently available tools. By model reduction we produce simpler models with less variables and parameters, that can be more easily simulated and analyzed. We propose a reduction method that applies to ordinary differential equations models of voltage and ligand gated ion channels coupled to signaling and metabolism. These models are used for studying various biological functions such as neuronal and cardiac activity, or insulin production by pancreatic beta-cells. Models of ion channels coupled to cell biochemistry share a common structure. For such models we identify fast and slow sub-processes, driving and slaved variables, as well as a set of reduced models. Various reduced models are valid locally and can change on a trajectory. The resulting reduction is hybrid, implying transitions from one reduced model (mode) to another one.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Institutional subscriptions

References

  1. Biktasheva, I., Simitev, R., Suckley, R., Biktashev, V.: Asymptotic properties of mathematical models of excitability. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 364(1842), 1283–1298 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bueno-Orovio, A., Cherry, E.M., Fenton, F.H.: Minimal model for human ventricular action potentials in tissue. J. Theoret. Biol. 253(3), 544–560 (2008)

    Article  MathSciNet  Google Scholar 

  3. Clewley, R.: Dominant-scale analysis for the automatic reduction of high-dimensional ODE systems. In: ICCS 2004 Proceedings, Complex Systems Institute, New England (2004)

    Google Scholar 

  4. Clewley, R., Rotstein, H.G., Kopell, N.: A computational tool for the reduction of nonlinear ODE systems possessing multiple scales. Multiscale Model. Simul. 4(3), 732–759 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fridlyand, L.E., Jacobson, D., Kuznetsov, A., Philipson, L.H.: A model of action potentials and fast Ca 2+ dynamics in pancreatic \(\beta \)-cells. Biophys. J. 96(8), 3126–3139 (2009)

    Article  Google Scholar 

  7. Fridlyand, L.E., Jacobson, D.A., Philipson, L.: Ion channels and regulation of insulin secretion in human \(\beta \)-cells: a computational systems analysis. Islets 5(1), 1–15 (2013)

    Article  Google Scholar 

  8. Fridlyand, L.E., Philipson, L.H.: Pancreatic beta cell G-protein coupled receptors and second messenger interactions: a systems biology computational analysis. PloS one 11(5), e0152869 (2016)

    Article  Google Scholar 

  9. Grosu, R., Batt, G., Fenton, F.H., Glimm, J., Le Guernic, C., Smolka, S.A., Bartocci, E.: From cardiac cells to genetic regulatory networks. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 396–411. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  10. Hille, B.: Ion Channels of Excitable Membranes. Sinauer, Sunderland (2001)

    Google Scholar 

  11. Hodgkin, A., Huxley, A.: Propagation of electrical signals along giant nerve fibres. Proc. R. Soc. Lond. Ser. B Biol. Sci. 140, 177–183 (1952)

    Article  Google Scholar 

  12. Holmes, M.H.: Introduction to Perturbation Methods, vol. 20. Springer Science & Business Media, New York (2012)

    Google Scholar 

  13. Iyer, V., Mazhari, R., Winslow, R.L.: A computational model of the human left-ventricular epicardial myocyte. Biophys. J. 87(3), 1507–1525 (2004)

    Article  Google Scholar 

  14. Keener, J.P.: Invariant manifold reductions for Markovian ion channel dynamics. J. Math. Biol. 58(3), 447–457 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Keener, J.P., Sneyd, J.: Mathematical Physiology, vol. 1. Springer, New York (1998)

    MATH  Google Scholar 

  16. Lagerstrom, P., Casten, R.: basic concepts underlying singular perturbation techniques. SIAM Rev. 14(1), 63–120 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. MATLAB: version 1.7.0_11 (R2013b). The MathWorks Inc., Natick, Massachusetts (2013)

    Google Scholar 

  18. Murthy, A., Islam, M.A., Bartocci, E., Cherry, E.M., Fenton, F.H., Glimm, J., Smolka, S.A., Grosu, R.: Approximate bisimulations for sodium channel dynamics. In: Gilbert, D., Heiner, M. (eds.) CMSB 2012. LNCS, vol. 7605, pp. 267–287. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  19. Noel, V., Grigoriev, D., Vakulenko, S., Radulescu, O.: Tropical geometries and dynamics of biochemical networks application to hybrid cell cycle models. Electron. Notes Theoret. Comput. Sci. 284, 75–91 (2012). In: Feret, J., Levchenko, A. (eds.) Proceedings of the 2nd International Workshop on Static Analysis and Systems Biology (SASB 2011). Elsevier

    Article  MathSciNet  MATH  Google Scholar 

  20. Noel, V., Grigoriev, D., Vakulenko, S., Radulescu, O.: Tropicalization and tropical equilibration of chemical reactions. In: Litvinov, G., Sergeev, S. (eds.) Tropical and Idempotent Mathematics and Applications, Contemporary Mathematics, vol. 616, pp. 261–277. American Mathematical Society (2014)

    Google Scholar 

  21. Radulescu, O., Gorban, A.N., Zinovyev, A., Noel, V.: Reduction of dynamical biochemical reactions networks in computational biology. Front. Genet. 3(131) (2012)

    Google Scholar 

  22. Radulescu, O., Swarup Samal, S., Naldi, A., Grigoriev, D., Weber, A.: Symbolic dynamics of biochemical pathways as finite states machines. In: Roux, O., Bourdon, J. (eds.) CMSB 2015. LNCS, vol. 9308, pp. 104–120. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  23. Radulescu, O., Vakulenko, S., Grigoriev, D.: Model reduction of biochemical reactions networks by tropical analysis methods. Math. Model Nat. Phenom. 10(3), 124–138 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Samal, S.S., Grigoriev, D., Fröhlich, H., Weber, A., Radulescu, O.: A geometric method for model reduction of biochemical networks with polynomial rate functions. Bull. Math. Biol. 77(12), 2180–2211 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Soliman, S., Fages, F., Radulescu, O.: A constraint solving approach to model reduction by tropical equilibration. Algorithms Mol. Biol. 9(1), 1 (2014)

    Article  Google Scholar 

  26. Suckley, R., Biktashev, V.N.: The asymptotic structure of the Hodgkin-Huxley equations. Int. J. Bifurcat. Chaos 13(12), 3805–3825 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tang, Y., Stephenson, J.L., Othmer, H.G.: Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics. Biophys. J. 70(1), 246 (1996)

    Article  Google Scholar 

  28. Tikhonov, A.N.: Systems of differential equations containing small parameters in the derivatives. Matematicheskii Sbornik 73(3), 575–586 (1952)

    MathSciNet  Google Scholar 

  29. Wechselberger, M., Mitry, J., Rinzel, J.: Canard theory and excitability. In: Kloeden, P.E., Pötzsche, C. (eds.) Nonautonomous Dynamical Systems in the Life Sciences. LIM, vol. 2102, pp. 89–132. Springer, Switzerland (2013)

    Chapter  Google Scholar 

Download references

Acknowledgments

This work was supported by the University of Chicago and by the FACCTS (France and Chicago Collaborating in The Sciences) program. The authors express their gratitude to the reviewers for their many helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ovidiu Radulescu .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Sommer-Simpson, J., Reinitz, J., Fridlyand, L., Philipson, L., Radulescu, O. (2016). Hybrid Reductions of Computational Models of Ion Channels Coupled to Cellular Biochemistry. In: Bartocci, E., Lio, P., Paoletti, N. (eds) Computational Methods in Systems Biology. CMSB 2016. Lecture Notes in Computer Science(), vol 9859. Springer, Cham. https://doi.org/10.1007/978-3-319-45177-0_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45177-0_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45176-3

  • Online ISBN: 978-3-319-45177-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics