Theory in Biosciences

, Volume 129, Issue 1, pp 25–38 | Cite as

Jensen’s inequality as a tool for explaining the effect of oscillations on the average cytosolic calcium concentration

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


It has often been asked which physiological advantages calcium (Ca2+) oscillations in non-excitable cells may have as compared to an adjustable stationary Ca2+ signal. One of the proposed answers is that an oscillatory regime allows a lowering of the average Ca2+ concentration, which is likely to be advantageous because Ca2+ is harmful to the cell in high concentrations. To check this hypothesis, we apply Jensen’s inequality to study the relation between the average Ca2+ concentration during oscillations and the Ca2+ concentration at the (unstable) steady state. Jensen’s inequality states that for a (strictly) convex function, the function value of the average of a set of argument values is lower than the average of the function values of the arguments from that set. We show that the kinetics of the Ca2+ efflux out of the cell is crucial in this context. By analytical calculations we derive that, if the Ca2+ efflux is a convex function of the cytosolic Ca2+ concentration, then oscillations lower the average Ca2+ concentration in comparison to the unstable steady state. If it is a concave function, the average Ca2+ concentration is increased, while it remains the same if that function is linear. We also analyse the case where the efflux obeys a Hill kinetics, which involves both a convex and a concave part. The results are illustrated by numerical simulations and simple example models. The theoretical predictions are tested with three experimental data sets from the literature. In two of them, the average appears to be higher than the steady-state value, while the third points to approximate equality. Thus oscillations may be used in real cells to tune the average Ca2+ concentration in both directions.


Calcium oscillations Jensen’s inequality Average calcium concentration Convex function Concave function Hill kinetics Advantage of oscillations 

List of symbols


Dissociation constant of CICR


Rate constant of pumping Ca2+ from the cytosol into the intracellular stores


Rate constant of CICR


Rate constant of the leak efflux


Half-saturation constant


Hill coefficient




Vout efflux function out of the cell


Influx function into the cell


ATPase pumping Ca2+ from the cytosol into the intracellular stores


Ca2+ release out of the ER following the CICR


Leak flux out of the intracellular stores


Overall concentration of Ca2+ in the intracellular stores


Concentration of Ca2+ in one of the intracellular stores


Concentration of Ca2+ in the cytosol


Steady-state concentration of Ca2+ in the cytosol

\( {\left\langle Z \right\rangle } \)

Average concentration of Ca2+ in the cytosol





Ca2+-induced Ca2+ release mechanism


Endoplasmic reticulum




Plasma membrane Ca2+ ATPase


Store-operated Ca2+ channels





Endoplasmic reticulum



We would like to thank E.G. Schukat-Talamazzini for support with the image analysis of time series and Ines Heiland for stimulating discussions. Financial support by the German Federal Ministry of Education and Research (BMBF) within the HepatoSys programme is gratefully acknowledged.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Beate Knoke
    • 1
    • 3
  • Christian Bodenstein
    • 1
  • Marko Marhl
    • 2
  • Matjaž Perc
    • 2
  • Stefan Schuster
    • 1
  1. 1.Department of BioinformaticsFriedrich Schiller University JenaJenaGermany
  2. 2.Department of Physics, Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  3. 3.Institute of Biochemical EngineeringUniversity of StuttgartStuttgartGermany

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