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Motivated by recent experimental studies, we derive and analyze atwodimensional model for the contraction patterns observed in protoplasmicdroplets of Physarum polycephalum. The model couples a model of an activeporoelastic two-phase medium with equations describing the spatiotemporaldynamics of the intracellular free calcium concentration. The poroelasticmedium is assumed to consist of an active viscoelastic solid representing thecytoskeleton and a viscous fluid describing the cytosol. The model equationsfor the poroelastic medium are obtained from continuum force-balance equationsthat include the relevant mechanical fields and an incompressibility relationfor the two-phase medium. The reaction-diffusion equations for the calciumdynamics in the protoplasm of Physarum are extended by advective transport dueto the flow of the cytosol generated by mechanical stresses. Moreover, weassume that the active tension in the solid cytoskeleton is regulated by thecalcium concentration in the fluid phase at the same location, which introducesa chemomechanical feedback. A linear stability analysis of the homogeneousstate without deformation and cytosolic flows exhibits an oscillatory Turinginstability for a large enough mechanochemical coupling strength. Numericalsimulations of the model equations reproduce a large variety of wave patterns,including traveling and standing waves, turbulent patterns, rotating spiralsand antiphase oscillations in line with experimental observations ofcontraction patterns in the protoplasmic droplets.

An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum