Date Published: August 9, 2019
Publisher: Public Library of Science
Author(s): Dirk Alexander Kulawiak, Jakob Löber, Markus Bär, Harald Engel, Fang-Bao Tian.
The onset of self-organized motion is studied in a poroelastic two-phase model with free boundaries for Physarum microplasmodia (MP). In the model, an active gel phase is assumed to be interpenetrated by a passive fluid phase on small length scales. A feedback loop between calcium kinetics, mechanical deformations, and induced fluid flow gives rise to pattern formation and the establishment of an axis of polarity. Altogether, we find that the calcium kinetics that breaks the conservation of the total calcium concentration in the model and a nonlinear friction between MP and substrate are both necessary ingredients to obtain an oscillatory movement with net motion of the MP. By numerical simulations in one spatial dimension, we find two different types of oscillations with net motion as well as modes with time-periodic or irregular switching of the axis of polarity. The more frequent type of net motion is characterized by mechano-chemical waves traveling from the front towards the rear. The second type is characterized by mechano-chemical waves that appear alternating from the front and the back. While both types exhibit oscillatory forward and backward movement with net motion in each cycle, the trajectory and gel flow pattern of the second type are also similar to recent experimental measurements of peristaltic MP motion. We found moving MPs in extended regions of experimentally accessible parameters, such as length, period and substrate friction strength. Simulations of the model show that the net speed increases with the length, provided that MPs are longer than a critical length of ≈ 120 μm. Both predictions are in line with recent experimental observations.
Dynamic processes in biological systems such as cells are examples of when spatio-temporal patterns develop far from thermodynamic equilibrium [1, 2]. One fascinating instance of such active matter are intracellular molecular motors that consume ATP  and can drive mechano-chemical contraction-expansion patterns  and, ultimately, cell locomotion. Further biological examples of such phenomena are discussed in [5–7].
We follow our earlier work [37–40, 48, 49] and utilize an active poroelastic two-phase model in one spatial dimension to describe homogeneous and isotropic Physarum microplasmodia (MP). We assume that MP consist of an active gel phase representing the cytoskeleton that we model as a viscoelastic solid with a displacement field u and velocity field u˙ . The gel is penetrated by a passive cytosolic fluid phase with flow velocity field v. For related models see [51, 52].
We solve Eq 6 (numerically in a one-dimensional domain of length L with parameters adopted from  which are listed in S1 Table, unless stated otherwise. Our initial condition is the weakly perturbed homogeneous steady state (HSS) with
In the present work, we have extended the two-phase model with free-boundaries for poroelastic droplets from  in two steps. First, we introduce a nonlinear slip-stick friction between droplet and substrate. While we were able to observe back and forth motion of the boundaries in , we showed that COM motion is impossible with a spatially homogeneous substrate friction. The nonlinear stick-slip approach results in a heterogeneous friction coefficient and allows for COM motion. Second, we consider a nonlinear oscillatory chemical kinetics that was derived for calcium as the regulator which controls the gel’s active tension in Physarum microplasmodia (MP) . It is known from previous studies [39, 40] that introducing a nonlinear calcium kinetics can lead to a uni-directional propagation of mechano-chemical waves that establishes an axis of polarity which is stable. This is in contrast to our previous work where the pure advection-diffusion dynamics of a passive regulator does not lead to the establishment of a polarity axis necessary for net motion .