  Motivated by recent experimental studies, we derive and analyze a
twodimensional model for the contraction patterns observed in protoplasmic
droplets of Physarum polycephalum. The model couples a model of an active
poroelastic two-phase medium with equations describing the spatiotemporal
dynamics of the intracellular free calcium concentration. The poroelastic
medium is assumed to consist of an active viscoelastic solid representing the
cytoskeleton and a viscous fluid describing the cytosol. The model equations
for the poroelastic medium are obtained from continuum force-balance equations
that include the relevant mechanical fields and an incompressibility relation
for the two-phase medium. The reaction-diffusion equations for the calcium
dynamics in the protoplasm of Physarum are extended by advective transport due
to the flow of the cytosol generated by mechanical stresses. Moreover, we
assume that the active tension in the solid cytoskeleton is regulated by the
calcium concentration in the fluid phase at the same location, which introduces
a chemomechanical feedback. A linear stability analysis of the homogeneous
state without deformation and cytosolic flows exhibits an oscillatory Turing
instability for a large enough mechanochemical coupling strength. Numerical
simulations of the model equations reproduce a large variety of wave patterns,
including traveling and standing waves, turbulent patterns, rotating spirals
and antiphase oscillations in line with experimental observations of
contraction patterns in the protoplasmic droplets.

Pattern Formation and Solitons;Cell Behavior