An Active Poroelastic Model for Mechanochemical Patterns in Protoplasmic Droplets of Physarum polycephalum

Abstract

Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description 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 equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition 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 stress. 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 mechanochemical coupling. 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.

Coordination of contractility, adhesion and flow in migratingPhysarumamoebae

“…Indeed, it is unclear how Physarum exerts stresses on its surroundings. Previous models have suggested that wave-like patterns of contraction may spontaneously arise from the coupling of the mechanics and chemistry of contraction in Physarum [20,21]. It is plausible that a similar mechanism may give rise to a wave-like modulation of the strength of adhesive interactions.…”

Mechanochemical pattern formation in simple models of active viscoelastic fluids and solids

“…In two-phase models of the cytoplasm suggested to model the dynamics of Physarum polycephalum the dynamics of the fluid viscous cytosol is taken into account in addition to the dynamics of the active solid viscoelastic cytoskeleton described by a Kelvin-Voigt model [20,21,23]. The resulting model equations are, however, apart from the sign of the active stress identical to the one-phase Kelvin-Voigt model given above and therefore, present, the best choice for modeling Physarum as an active solid.…”

“…Radszuweit et al [20] showed that the whole variety of deformations pattern observed in experiments with plasmodial droplets of Physarum polycephalum reported in [8,9,10,11] require also an extension of the active poroelastic model by reactiondiffusion equations describing the regulation of calcium ions. …”

“…A reduction of detailed models of the cytoplasm of Physarum [19,20,21] gives rise to a simple and more generic model of mechanochemical pattern formation in active viscoelastic cytoskeleton [23]. Since contraction and the subsequent deformations in Physarum were correlated with low calcium concentration [9], the cited models employ an inhibitory regulator for the active stress in the cytoskeleton.…”

Oscillations and uniaxial mechanochemical waves in a model of an active poroelastic medium: Application to deformation patterns in protoplasmic droplets of Physarum polycephalum

“…The concept of Physarum cells as being poroelastic media was used previously in a series of studies aiming at understanding the dynamics of the reorganization of protoplasmic droplets and the related deformation patterns [3,44,45].…”