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.
Highlights
The true slime mold Physarum polycephalum is an extensively studied system in biophysics
In this context the term ‘‘intelligent’’ means that, given an external stimulus, the plasmodium optimizes its cell shape, vein network and growth with respect to transport efficiency as well as robustness with respect to link deletion and avoidance of unfavorable conditions [2,3]. Recent experiments along these lines show that plasmodia were able to reproduce public transport networks on the scale of a petri dish [4] and to ‘‘solve’’ maze problems such as finding the shortest path between two food sources placed at the exits of a labyrinth [5]
A second remarkable phenomenon is the synchronization of the contraction patterns in the tubular vein network that generates shuttle streaming to distribute nutrients efficiently throughout the organism [9]
Summary
The true slime mold Physarum polycephalum is an extensively studied system in biophysics. The plasmodial stage is of particular interest, since it exhibits, despite the relatively simple organization of this unicellular organism, seemingly ‘‘intelligent’’ physiological processes [1] In this context the term ‘‘intelligent’’ means that, given an external stimulus, the plasmodium optimizes its cell shape, vein network and growth with respect to transport efficiency as well as robustness with respect to link deletion and avoidance of unfavorable conditions [2,3]. Recent experiments along these lines show that plasmodia were able to reproduce public transport networks on the scale of a petri dish [4] and to ‘‘solve’’ maze problems such as finding the shortest path between two food sources placed at the exits of a labyrinth [5]. For the formulation of mathematical models a basic understanding of chemical and mechanical processes in the protoplasm is needed
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