# Annihilation of calcium oscillations

Annihilation of calcium oscillations | |
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State dimension: | 1 |

Differential states: | 4 |

Discrete control functions: | 1 |

Path constraints: | 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.

## Contents

## 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 almost everywhere the mixed-integer optimal control problem is given by

Here the differential states 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 [Kummer2000]**Author: ** *U. Kummer; L.F. Olsen; C.J. Dixon; A.K. Green; E. Bornberg-Bauer; G. Baier***Journal: ** *Biophysical Journal***Month: ** *September***Pages: ** *1188--1195***Title: ** *Switching from Simple to Complex Oscillations in Calcium Signaling***Volume: ** *79***Year: ** *2000*

. In the given equations that stem from [Lebiedz2005]**Author: ** *Lebiedz, D.; Sager, S.; Bock, H.G.; Lebiedz, P.***Journal: ** *Physical Review Letters***Pages: ** *108303***Title: ** *Annihilation of limit cycle oscillations by identification of critical phase resetting stimuli via mixed-integer optimal control methods***Volume: ** *95***Year: ** *2005*

, the model is identical to the one derived there, except for an additional first-order leakage flow of calcium from the ER back to the cytoplasm, which is modeled by in equations 3 and 4. It reproduces well experimental observations of cytoplasmic calcium oscillations as well as bursting behavior and in particular the frequency encoding of the triggering stimulus strength, which is a well known mechanism for signal processing in cell biology.

## Parameters

These fixed values are used within the model.

## Reference Solutions

The depicted optimal solution consists of a stimulus of and a timing given by the stage lengths 4.6947115, 0.1491038, and 17.1561845. The optimal objective function value is 1610.654. As can be seen from the additional plots, this solution is extremely unstable. A small perturbation in the control, or simply rounding errors on a longer time horizon lead to a transition back to the stable limit-cycle oscillations.

## Optimization

The determination of the stimulus by means of optimization is quite hard for two reasons. First, the unstable target steady-state. Only a stable all-at-once algorithm such as multiple shooting or collocation can be applied successfully.

Second, the objective landscape of the problem in switching time formulation (this is, for a fixed stimulus strength and modifying only beginning and length of the stimulus) is quite nasty, as the following visualizations obtained by simulation show.

## Source Code

- Muscod code at Annihilation of calcium oscillations (Muscod)
- JModelica code at Annihilation of calcium oscillations (jModelica)

## Variants

- Use of two control functions.
- Scaled deviation in objective function.

## References

[Kummer2000] | U. Kummer; L.F. Olsen; C.J. Dixon; A.K. Green; E. Bornberg-Bauer; G. Baier (2000): Switching from Simple to Complex Oscillations in Calcium Signaling. Biophysical Journal, 79, 1188--1195 | |

[Lebiedz2005] | Lebiedz, D.; Sager, S.; Bock, H.G.; Lebiedz, P. (2005): Annihilation of limit cycle oscillations by identification of critical phase resetting stimuli via mixed-integer optimal control methods. Physical Review Letters, 95, 108303 |