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

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.