Annihilation of calcium oscillations

Annihilation of calcium oscillations
State dimension:	1
Differential states:	4
Discrete control functions:	1
Interior point equalities:	4

The aim of the control problem is to identify strength and timing of inhibitor stimuli that lead to a phase singularity which annihilates intracellular calcium oscillations. This is formulated as an objective function that aims at minimizing the state deviation from a desired unstable steady state, integrated over time.

The mathematical equations form a small-scale ODE model. The interior point equality conditions fix the initial values of the differential states. The problem is, despite of its low dimension, very hard to solve, as the target state is unstable.

Biological motivation

Biological rhythms as impressing manifestations of self-organized dynamics associated with the phenomenon life have been of particular interest since quite a long time. Even before the mechanistic basis of certain biochemical oscillators was elucidated by molecular biology techniques, their investigation and the issue of perturbation by external stimuli has attracted much attention.

A calcium oscillator model describing intracellular calcium spiking in hepatocytes induced by an extracellular increase in adenosine triphosphate (ATP) concentration is described. The calcium signaling pathway is initiated via a receptor activated G-protein inducing the intracellular release of inositol triphoshate (IP3) by phospholipase C. The IP3 triggers the opening of endoplasmic reticulum and plasma membrane calcium channels and a subsequent inflow of calcium ions from intracellular and extracellular stores leading to transient calcium spikes.

As a source of external control a temporally varying concentration of an uncompetitive inhibitor of the PMCA ion pump is considered.

Mathematical formulation

For $t \in [t_0, t_f]$ almost everywhere the mixed-integer optimal control problem is given by

$\begin{array}{llcl} \displaystyle \min_{x, w, w^{\mathrm{max}}} & & & {\int_{t_0}^{t_f} || x(\tau) - \tilde{x} ||_2^2 + p_1 w(\tau) \; \mathrm{d}\tau} \\[1.5ex] \mbox{s.t.} & \dot{x}_0 & = & k_1 + k_2 x_0 - \frac{k_3 x_0 x_1}{x_0 + K_4} - \frac{k_5 x_0 x_2}{x_0 + K_6} \\ & \dot{x}_1 & = & k_7 x_0 - \frac{k_8 x_1}{x_1 + K_9} \\ & \dot{x}_2 & = & \frac{k_{10} x_1 x_2 x_3}{x_3 + K_{11}} + k_{12} x_1 + k_{13} x_0 - \frac{k_{14} x_2}{w \cdot x_2 + K_{15}} - \frac{k_{16} x_2}{x_2 + K_{17}} + \frac{x_3}{10} \\ & \dot{x}_3 & = & - \frac{k_{10} x_1 x_2 x_3}{x_3 + K_{11}} + \frac{k_{16} x_2}{x_2 + K_{17}} - \frac{x_3}{10} \\[1.5ex] & x(0) &=& (0.03966, 1.09799, 0.00142, 1.65431)^T, \\ & x(t) & \ge & 0.0, \\ & w(t) &\in& \{1, w^{\mathrm{max}}\}, \\ & w^{\mathrm{max}} & \ge & 1.1, \\ & w^{\mathrm{max}} & \le & 1.3. \end{array}$

Here the differential states $(x_0, x_1, x_2, x_3)$ describe concentrations of activated G-proteins, active phospholipase C, intracellular calcium, and intra-ER calcium, respectively.

Modeling details including a comprehensive discussion of parameter values and the dynamical behavior observed in simulations with a comparison to experimental observations can be found in \[Citationkey missing]