Abstract
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.
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Abbreviations
- K :
-
Dissociation constant of CICR
- k 2 :
-
Rate constant of pumping Ca2+ from the cytosol into the intracellular stores
- k 3 :
-
Rate constant of CICR
- k f :
-
Rate constant of the leak efflux
- K S :
-
Half-saturation constant
- n :
-
Hill coefficient
- t :
-
Time
- f(Z):
-
Vout efflux function out of the cell
- V in :
-
Influx function into the cell
- V pump :
-
ATPase pumping Ca2+ from the cytosol into the intracellular stores
- V CICR :
-
Ca2+ release out of the ER following the CICR
- V leak :
-
Leak flux out of the intracellular stores
- Y :
-
Overall concentration of Ca2+ in the intracellular stores
- Y i :
-
Concentration of Ca2+ in one of the intracellular stores
- Z :
-
Concentration of Ca2+ in the cytosol
- Z ss :
-
Steady-state concentration of Ca2+ in the cytosol
- \( {\left\langle Z \right\rangle } \) :
-
Average concentration of Ca2+ in the cytosol
- CaM:
-
Calmodulin
- CICR:
-
Ca2+-induced Ca2+ release mechanism
- ER:
-
Endoplasmic reticulum
- IP3 :
-
Inositol-1,4,5-trisphosphate
- PMCA:
-
Plasma membrane Ca2+ ATPase
- SOCs:
-
Store-operated Ca2+ channels
- m:
-
Mitochondria
- ER:
-
Endoplasmic reticulum
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Acknowledgements
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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Appendix A: Analysis of the square-shaped signal with arbitrary baseline
Appendix A: Analysis of the square-shaped signal with arbitrary baseline
The general square-shaped Ca2+ signal is given by:
with the baseline Z 1 < Z 2. We can compute the average of Z(t) over one period T as:
where we have introduced the relative baseline 0 ≤ α = Z 1/Z 2 < 1. Next we calculate the average of f(Z(t)) (f being the Hill kinetics from Eq. 21) over one period:
From relations 13, 34 and 35, it follows:
Factoring out Z 2 on both sides and introducing the relative saturation μ = Z 2/K S, we can write:
This can be solved for μ:
When plotting the allowed range of μ (the area below μcrit) over γ as in Fig. 4, we can see that it decreases with increasing α (see Fig. 7 with n = 2). This corresponds to increasing the baseline concentration of Ca2+.
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Knoke, B., Bodenstein, C., Marhl, M. et al. Jensen’s inequality as a tool for explaining the effect of oscillations on the average cytosolic calcium concentration. Theory Biosci. 129, 25–38 (2010). https://doi.org/10.1007/s12064-010-0080-1
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DOI: https://doi.org/10.1007/s12064-010-0080-1