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 (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.

Appendix A: Analysis of the square-shaped signal with arbitrary baseline

The general square-shaped Ca²⁺ signal is given by:

$$ {Z (t )= \left\{ {\begin{array}{*{20}c} {Z_{ 1} ,} {t\bmod T > T_{\text{p}} } \\ {Z_{2} ,} {t\bmod T \le T} \\ \end{array} } \right.}, $$

(33)

with the baseline Z ₁ < Z ₂. We can compute the average of Z(t) over one period T as:

$$ {\left\langle Z \right\rangle = {\frac{1}{T}}\int\limits_{0}^{T} {Z(t){\text{d}}t} = \gamma Z_{ 2} + \left( {1 - \gamma } \right)}Z_{1} = Z_{2} \left( {\gamma + \left( {1 - \gamma } \right)\alpha } \right), $$

(34)

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:

$$ {\left\langle {f\left( Z \right)} \right\rangle = {\frac{1}{T}}\int\limits_{0}^{T} {f\left( {Z (t )} \right){\text{d}}t} = V_{\max } \left[ {\gamma {\frac{{Z_{2}^{n} }}{{K_{\text{S}}^{n} + Z_{2}^{n} }}} + (1 - \gamma ){\frac{{\left( {\alpha Z_{2} } \right)^{n} }}{{K_{\text{S}}^{n} + \left( {\alpha Z_{2} } \right)^{n} }}}} \right]}. $$

(35)

From relations 13, 34 and 35, it follows:

$$ {\frac{{Z_{2}^{n} \left( {\gamma + \left( {1 - \gamma } \right)\alpha } \right)^{n} }}{{K_{\text{S}}^{n} + Z_{2}^{n} \left( {\gamma + \left( {1 - \gamma } \right)\alpha } \right)^{n} }}}\;\le\;\gamma {\frac{{Z_{2}^{n} }}{{K_{\text{S}}^{n} + Z_{2}^{n} }}} + (1 - \gamma ){\frac{{\left( {\alpha Z_{2} } \right)^{n} }}{{K_{\text{S}}^{n} + \left( {\alpha Z_{2} } \right)^{n} }}}. $$

(36)

Factoring out Z ₂ on both sides and introducing the relative saturation μ = Z ₂/K _S, we can write:

$$ {\frac{{\left( {\gamma + \left( {1 - \gamma } \right)\alpha } \right)^{n} }}{{{\frac{1}{{\mu^{n} }}} + \left( {\gamma + \left( {1 - \gamma } \right)\alpha } \right)^{n} }}}\;\le\;\gamma {\frac{1}{{{\frac{1}{{\mu^{n} }}} + 1}}} + (1 - \gamma ){\frac{{\alpha^{n} }}{{{\frac{1}{{\mu^{n} }}} + \alpha^{n} }}}. $$

(37)

This can be solved for μ:

$$ \mu \le \mu_{\text{crit}} = \left( {{\frac{{\alpha^{n} + \gamma \left( {1 - \alpha^{n} } \right) - \left( {\alpha + \gamma - \alpha \gamma } \right)^{n} }}{{\left( {1 - \gamma \left( {1 - \alpha^{n} } \right)} \right)\left( {\alpha + \gamma - \alpha \gamma } \right)^{n} - \alpha^{n} }}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}}} . $$

(38)

When plotting the allowed range of μ (the area below μ_crit) 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⁺.