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

Abstract

It has often been asked which physiological advantages calcium (Ca(2+)) oscillations in non-excitable cells may have as compared to an adjustable stationary Ca(2+) signal. One of the proposed answers is that an oscillatory regime allows a lowering of the average Ca(2+) concentration, which is likely to be advantageous because Ca(2+) is harmful to the cell in high concentrations. To check this hypothesis, we apply Jensen's inequality to study the relation between the average Ca(2+) concentration during oscillations and the Ca(2+) 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 Ca(2+) efflux out of the cell is crucial in this context. By analytical calculations we derive that, if the Ca(2+) efflux is a convex function of the cytosolic Ca(2+) concentration, then oscillations lower the average Ca(2+) concentration in comparison to the unstable steady state. If it is a concave function, the average Ca(2+) 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 Ca(2+) concentration in both directions.

Robust Concentration and Frequency Control in Oscillatory Homeostats

“…Although a variety of mathematical models have been suggested to describe Ca 2+ oscillations [78]–[84], none of them have so far included an explicit homeostatic regulation of cytosolic Ca 2+ . Fig.…”

Differential equation-based minimal model describing metabolic oscillations in Bacillus subtilis biofilms

“…It is of interest to speculate about the physiological advantage of spike-like oscillations. This question has been discussed earlier in the context of calcium oscillations [26,38,39]. Whenever the kinetic effect of the oscillating variable (e.g.…”

“…To study the effect and possible benefit of oscillations, it is of interest to compute the average values of variables, as was done for several oscillators [24][25][26][27][28][29]. For linear differential equation systems showing oscillations (such as the system describing the harmonic pendulum), the average values equal the values at the marginally stable steady state.…”

Differential equation based minimal model describing metabolic oscillations inBacillus subtilisbiofilms

“…It is of interest to speculate about the physiological advantage of spike-like oscillations. This question has been discussed earlier in the context of calcium oscillations 20,30,31 Whenever the kinetic effect of the oscillating variable (e.g. in activation of a protein or in a biochemical conversion) is nonlinear and follows a convex function, the spikes contribute more than proportionately to the effect.…”