Stability in generic mitochondrial models

  • Pete Donnell
  • Murad Banaji
  • Stephen BaigentEmail author
Original Paper


In this paper, we use a variety of mathematical techniques to explore existence, local stability, and global stability of equilibria in abstract models of mitochondrial metabolism. The class of models constructed is defined by the biochemical description of the system—an electron transport chain coupled to a process of charge translocation across a membrane. The conclusions are based on the reaction network structure, and we make minimal assumptions on the kinetics of the reactions involved. In the absence of charge translocation these models have previously been shown to behave in a very simple manner with a single, globally stable equilibrium. We show that with charge translocation the conclusion about a unique equilibrium remains true, but local and global stability do not necessarily follow. The length of the chains proves to be important: For short electron transport chains it is possible to make claims about local and global stability of the equilibrium which are no longer valid for longer chains. Some particular conditions which ensure stability of the equilibrium for chains of arbitrary length are presented.


Mitochondria Electron transport Stability Logarithmic norms 


  1. 1.
    Babcock G.T., Wikström M.: Oxygen activation and the conservation of energy in cell respiration. Nature 356, 301 (1992)CrossRefGoogle Scholar
  2. 2.
    R.H. Garrett, C.M. Grisham (eds.), Biochemistry (Saunders College Publishing, 1995)Google Scholar
  3. 3.
    N. Bhagavan, Medical Biochemistry (Harcourt/Academic Press, 2002)Google Scholar
  4. 4.
    D.G. Nicholls, S.J. Ferguson, Bioenergetics 3 (Academic Press, 2002)Google Scholar
  5. 5.
    Belevich I., Verkhovsky M., Wikström M.: Proton-coupled electron transfer drives the proton pump of cytochrome c oxidase. Nature 440(6), 829 (2006)CrossRefGoogle Scholar
  6. 6.
    Banaji M.: A generic model of electron transport in mitochondria. J. Theor. Biol. 243(4), 501 (2006)CrossRefGoogle Scholar
  7. 7.
    Banaji M., Baigent S.: Electron transfer networks. J. Math. Chem. 43(4), 1355 (2008)CrossRefGoogle Scholar
  8. 8.
    Korzeniewski B.: Simulation of oxidative phosphorylation in hepatocytes. Biophys. Chem. 58, 215 (1996)CrossRefGoogle Scholar
  9. 9.
    Korzeniewski B., Zoladz J.A.: A model of oxidative phosphorylation in mammalian skeletal muscle. Biophys. Chem. 92, 17 (2001)CrossRefGoogle Scholar
  10. 10.
    Farmery A.D., Whiteley J.P.: A mathematical model of electron transfer within the mitochondrial respiratory cytochromes. J. Theor. Biol. 213, 197 (2001)CrossRefGoogle Scholar
  11. 11.
    Beard D.A.: A biophysical model of the mitochondrial respiratory system and oxidative phosphorylation. PLoS Comput. Biol. 1(4), e36 (2005)CrossRefGoogle Scholar
  12. 12.
    Jin Q., Bethke C.M.: Kinetics of electron transfer through the respiratory chain. Biophys. J. 83(4), 1797 (2002)CrossRefGoogle Scholar
  13. 13.
    De Leenheer P., Angeli D., Sontag E.: Monotone chemical reaction networks. J. Math. Chem. 41(3), 295 (2007)CrossRefGoogle Scholar
  14. 14.
    Brand M.D., Chien L., Diolez P.: Experimental discrimination between proton leak and redox slip during mitochondrial electron transport. Biochem. J. 297(1), 27 (1994)Google Scholar
  15. 15.
    Canton M., Luvisetto S., Schmehl I., Azzone G.: The nature of mitochondrial respiration and discrimination between membrane and pump properties. Biochem. J. 310, 477 (1995)Google Scholar
  16. 16.
    Banaji M., Donnell P., Baigent S.: P matrix properties, injectivity and stability in chemical reaction systems. SIAM J. Appl. Math. 67(6), 1523 (2007)CrossRefGoogle Scholar
  17. 17.
    Kellogg R.B.: On complex eigenvalues of M and P matrices. Numer. Math. 19, 70 (1972)CrossRefGoogle Scholar
  18. 18.
    Gale D., Nikaido H.: The Jacobian matrix and global univalence of mappings. Math. Ann. 159, 81 (1965)CrossRefGoogle Scholar
  19. 19.
    Feßler R.: A proof of the two-dimensional Markus-Yamabe stability conjecture. Ann. Polon. Math. 62, 45 (1995)Google Scholar
  20. 20.
    Glutsyuk A.A.: The complete solution of the Jacobian problem for vector fields on the plane. Russ. Math. Surv. 49(3), 185 (1994)CrossRefGoogle Scholar
  21. 21.
    Gutierrez C.: A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, 627 (1995)Google Scholar
  22. 22.
    K. Ciesielski, On the Poincaré-Bendixson theorem. in Lecture Notes in Nonlinear Analysis, vol 3, eds. by W. Kryszewski, A. Nowakowski, Proceedings of the 3rd Polish Symposium on Nonlinear Analysis (2001)Google Scholar
  23. 23.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983)Google Scholar
  24. 24.
    Cima A., Essen A.V., Gasull A., Hubbers E., Manosas F.: A polynomial counterexample to the Markus-Yamabe conjecture. Adv. Math. 131(2), 453 (1997)CrossRefGoogle Scholar
  25. 25.
    Muldowney J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20(4), 857 (1990)CrossRefGoogle Scholar
  26. 26.
    Li M.Y., Wang L.: A criterion for stability of matrices. J. Math. Anal. Appl. 225, 249 (1998)CrossRefGoogle Scholar
  27. 27.
    Scilab, A platform for numerical computation. Available at
  28. 28.
    Kafri W.: Robust D-stability. Appl. Math. Lett. 15, 7 (2002)CrossRefGoogle Scholar
  29. 29.
    M. Hirsch, H. Smith, Monotone dynamical systems. in Handbook of Differential Equations: Ordinary Differential Equations, vol 2, eds. by A. Canada, P. Drabek, A. Fonda (Elsevier, 2005)Google Scholar
  30. 30.
    D. Angeli, E. Sontag, Interconnections of monotone systems with steady-state characteristics. in Optimal Control, Stabilization and Nonsmooth Analysis, eds. by M.M.M. de Queiroz, P. Wolenski (Springer-Verlag, 2004)Google Scholar
  31. 31.
    Angeli D., Sontag E.: Monotone control systems. IEEE Trans. Automat. Contr. 48, 1684 (2003)CrossRefGoogle Scholar
  32. 32.
    Li M.Y., Muldowney J.S.: Dynamics of differential equations on invariant manifolds. J. Differ. Equ. 168, 295 (2000)CrossRefGoogle Scholar
  33. 33.
    L. Allen, T.J. Bridges, Numerical exterior algebra and the compound matrix method. Tech. rep. University of Surrey (2001)Google Scholar
  34. 34.
    Ström T.: On logarithmic norms. SIAM J. Numer. Anal. 12(5), 741 (1975)CrossRefGoogle Scholar
  35. 35.
    MAXIMA: A computer algebra system. Available at

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Medical Physics and BioengineeringUniversity College LondonLondonUK
  2. 2.Department of MathematicsUniversity College LondonLondonUK

Personalised recommendations