Theory in Biosciences

, Volume 127, Issue 1, pp 1–14 | Cite as

Equality of average and steady-state levels in some nonlinear models of biological oscillations

  • Beate Knoke
  • Marko Marhl
  • Matjaž Perc
  • Stefan Schuster
Original Paper


Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis.


Calcium oscillations Chaotic dynamics Glycolytic oscillations Lotka–Volterra equations Metabolic control analysis 



Travel grants from the Slovenian and German Ministries of Research and Education (grant nos. BI-DE/03-04-003 and SVN 02/013, respectively) and from the E.U. within the SOCRATES program for mutual working and teaching visits are gratefully acknowledged. We thank Gunter Neumann for drawing our attention to mathematical properties of Lotka–Volterra systems and Reinhart Heinrich, Thomas Höfer, Ursula Kummer, Jörg Stucki and Jana Wolf for stimulating discussions. We are grateful to Ina Weiß for valuable assistance in the literature search.


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Beate Knoke
    • 1
  • Marko Marhl
    • 2
  • Matjaž Perc
    • 2
  • Stefan Schuster
    • 1
  1. 1.Department of Bioinformatics, Faculty of Biology and PharmaceuticsFriedrich Schiller University of JenaJenaGermany
  2. 2.Department of Physics, Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia

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