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.

Figure 1. Contractions patterns.

Experiments with protoplasmic droplets of Physarum polycephalum , . The color represents the local phase of oscillation obtained by a Fourier transformation of the spatiotemporal height data: a) standing wave, b) many irregular spirals, c) traveling wave, d) antiphase patterns, and e) single spiral.

Figure 2. Schematic representation of the the two-phase model.

Drawing of a Physarum microdroplet (top) in side view showing the plasmalemma invaginations filled with extracellular matrix (light blue), the fluid phase of the cytoplasm (blue) and the solid filamentous phase (black). Top view of the droplet in the simplified framework of our two-phase model (bottom). Deformations of the cytoskeleton are represented by a distorted grid, flow field in the cytosol by arrows and free concentration in yellow, respectively.

Figure 3. Linear dispersion relation for varying mechanical coupling strength.

Real part (left) and imaginary part (right) of the branch of the dispersion relation with largest real parts of the eigenvalues for a) a stable HSS ( ) and b) an unstable HSS ( ). The imaginary part is nonzero for all unstable wavenumbers: . The mechanochemical coupling strength is varied in each case a) and b) increasing from dark to light blue. The drag coefficient is chosen to be and the remaining parameters are given in Tables 1 and 2.

Figure 4. Linear dispersion relation for varying drag coefficient.

Branch of the dispersion relation with largest real part a) for different drag coefficients that range over three orders of magnitude (logarithmic scale, from dark blue to cyan). The mechanochemical coupling strength is kept constant at . b) Wavelength of the fastest growing mode versus drag coefficient for three different mechanochemical coupling strengths (solid line), (dashed line) and (dotted line). The remaining parameters are given in Tables 1 and 2.

Figure 5. Phase diagram.

Phase diagram in the plane spanned by the mechanochemical coupling strength and the drag coefficient . The black line denotes the threshold coupling strength , where the most unstable mode according to linear stability analysis of the HSS is nonzero. The dashed blue curve separates the plane according to Eq. 20 in regions with (larger ) and (smaller ).

Figure 6. Traveling wave.

a) Snapshot of the free calcium concentration and c) relative height field in color and the protoplasmic flow field shown by arrows with length . Space-time plot of b) and d) along the dotted line in panel a). The period of local oscillations is . The parameters are and . The remaining values can be found in Tables 1 and 2. A video file displaying the spatiotemporal dynamics including the transient initial phase corresponding to subfigure c) is included in the supporting material (see Movie S1 in File S1).

Figure 7. Standing wave.

a) Snapshot of the free calcium concentration and c) relative height field in color and the protoplasmic flow field shown by arrows with length . Space-time plot of b) and d) along the dotted line in panel a). The period of local oscillations is . The parameters are and . The remaining values can be found in Tables 1 and 2. A video file corresponding to subfigure c) is included in the supporting material (see Movie S2 in File S1).

Figure 8. Single rotating spiral.

a) Snapshot of the free calcium concentration and c) relative height field in color and the protoplasmic flow field shown by arrows with length . Space-time plot of b) and d) along a circle marked by the dotted line in a). The period of local oscillations is . The parameters are and . For the remaining values see Tables 1 and 2. A video file corresponding to subfigure c) is included in the supporting material (see Movie S3 in File S1).

Figure 9. Radial wave.

a) Snapshot of the free calcium concentration and c) relative height field in color and the protoplasmic flow field shown by arrows with length . Space-time plot of b) and d) along the dotted line in a). The period of local oscillations is . The parameters are and . For the remaining values see Tables 1 and 2. A video file corresponding to subfigure c) is included in the supporting material (see Movie S4 in File S1).

Figure 10. Irregular wave pattern.

a) Snapshot of the free calcium concentration and c) relative height field in color and the protoplasmic flow field shown by arrows with length . Space-time plot of b) and d) along the dotted line in a). The parameters are and . For remaining values see Tables 1 and 2. A video file corresponding to subfigure c) is included in the supporting material (see Movie S5 in File S1).