Active poroelastic two-phase model for the motion of physarum microplasmodia
Active poroelastic two-phase model for the motion of physarum microplasmodia
Dirk Alexander KulawiakID 0 1 2
Jakob L o?ber 0 1 2
Markus B a?r 0 1 2
Harald Engel 0 1 2
0 Editor: Fang-Bao Tian, University of New South Wales , AUSTRALIA
1 1 TU Berlin - Institut f u ?r Theoretische Physik , Berlin, Germany , 2 Max-Planck-Institut f u ?r Physik komplexer Systeme , Dresden, Germany, 3 Physikalisch-Technische Bundesanstalt, Berlin , Germany
2 Funding: DAK was funded by the German Science Foundation (DFG) within the GRK 1558 and the collaborative research center SFB 910. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript
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 ?m. Both predictions are in line with recent experimental observations.
Competing interests: The authors have declared
that no competing interests exist.
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].
The true slime mold Physarum polycephalum is a well known model organism  that
exhibits mechano-chemical spatio-temporal patterns. Previous research in Physarum has
addressed many different topics in biophysics, such as genetic activity , habituation ,
decision making  and cell locomotion [12, 13]. Physarum is an unicellular organism,
which builds large networks that exhibit self-organized synchronized contraction patterns [8,
14, 15]. These contractions enable shuttle streaming in the tubular veins of the network and
allow for efficient nutrient transport throughout the organism . Many groups have
investigated the network?s dynamics [17?19], however size and complex topology of these networks
make analyzing and modeling them challenging.
Physarum microplasmodia (MP) allow one to study Physarum?s internal dynamics in a
simpler setup. These MPs can be produced by extracting cytoplasm from a vein and placing it
on a substrate. After reorganization, these droplets of cytoplasm show a surprising wealth of
spatio-temporal patterns such as spiral, standing and traveling waves and irregular and
antiphase oscillations in their height . After several hours, MPs are deforming to a tadpole-like
shape and start exploring their surroundings [21?24].
MP motion is composed of an oscillatory forward and backward motion. Their forward
motion has a larger magnitude than the backward motion. There are experimental
observations of two distinct motility types: peristaltic and amphistaltic [13, 24]. In the more common
peristaltic case forward traveling contraction waves result in forward motion of the whole cell
body. In the amphistaltic type, standing waves cause front and rear to contract in anti-phase.
Different modes of motion driven by periodic deformation waves are not restricted to
Physarum MP, but are also found in rather general mathematical models  and in the
locomotion of a wide variety of limbless and legged animals . More recently, different modes have
also been found in the Belousov-Zhabotinsky reaction in a light-driven photosensitive gel [27?
30]; Epstein and coworkers have classified the modes using the terms retrograde and direct
wave locomotion  and also report on a form of oscillatory migration without net
displacement of the average position .
Common models for cell locomotion and the cytoskeleton?s dynamics are based on active
fluid and gel models [31?34]. While simple fluids and solids are governed by a single
momentum balance equation, poroelastic media belong to the class of two-fluid models and possess
individual momentum balance equations for each of the phases. This approach is useful if the
two constituent phases have largely different rheological properties and penetrate each other
on the relatively small length scales on which cytosol permeates the cytoskeleton .
To describe the MP?s motion, we utilize an active poroelastic two-phase description [35,
36]. Active poroelastic models have been used to describe the pattern formation in resting 
and moving  poroelastic droplets. In simple generic models, a feedback loop between a
chemical regulator, mechanical contractions and induced flows give rise to pattern formation
in a resting droplet . In , we have studied this model with free boundary conditions
and linear friction between droplet and substrate. While we were able to observe back and
forth motion of the boundaries, the center of mass (COM) position remained fixed. The
droplet did not exhibit net motion. Moreover, we derived an argument that COM motion is
impossible with a spatially homogeneous substrate friction. Furthermore, numerical simulations
have shown that an additional mechanism establishing an axis of polarity that is stable on long
timescales is necessary for net motion .
In this work, we proceed in two steps. First we extend the model described in  with a
nonlinear slip-stick friction between droplet and substrate to create a spatially heterogeneous
substrate friction. Second, we consider a specific model derived for Physarum MP that
contains a reaction kinetics. Within this enhanced model, we explore the conditions for the onset
of motion of Physarum MP.
Previous work on models for resting Physarum MP by Radszuweit et al. has shown that
inclusion of a nonlinear reaction kinetics for the calcium regulator can result in the emergence
of uni-directional traveling mechano-chemical waves that establish an axis of polarity inside
the MP [39, 40]. Nonlinear friction is a common assumption in biology [41, 42] and can also
be found in other eucaryotic cells [43, 44]. Here, we utilize a simplified version of the nonlinear
slip-stick friction model introduced by Barnhart et al. to account for oscillatory modulations
of keratocyte movement . The nonlinear friction dynamics and its synchronization with
other properties such as the local strain inside a cell are also important for other types of
amoeboid motion such as chimneying . Moreover, regulation of the friction strength is
significant for locomotion in other biological systems such as snails or slugs [46, 47].
Section 2 contains a comprehensive description of the model. In Section 3, we analyze the
linear stability of the model and identify different types of motion. Furthermore, we explore
the effects of parameters that are accessible in the experiment such as the period of the internal
dynamics, the length and the substrate friction strength. Section 3 contains a comparison of
our findings with recent experiments of Physarum MP. Afterwards, we summarize our results
in the discussion and briefly address possible extensions of the model.
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
The total stress in the medium is given by ? = ?g?g + ?f ?f, where ?g (?f) denotes the volume
fraction and ?g (?f) the stress tensor in gel and fluid phase, respectively. While the time
evolution for the gel fraction is given by r^g ? rg?1 @xu?, only the constant term ?g enters into the
equations due to our small strain approximation with |@xu| 1. Assuming that no other
phases are present, the volume fractions obey ?g + ?f = 1 .
Each phase satisfies a momentum balance equation @x(?g/f ? p) + fg/f + fsub = 0, where p
denotes the hydrodynamic pressure. The friction between both phases is given by Darcy?s law
with fg ? ff ? rgrf b?v u_ ?. Following , we assume that the friction between gel and
substrate depends nonlinearly on the the local gel speed according to fsub ? g?ju_ j?u_ and no
friction between fluid and substrate.
We model MPs in an one-dimensional time-dependent domain B with boundaries denoted
by @B. Assuming that B is infinitely large in the y-direction, the boundary is straight, and we
omit terms that depend on interface tension or bending. Free boundary conditions in
x-direction enable boundary deformations and thus motion in response to bulk deformations [54?
56]. The total stress has to be continuous across the boundary and with the assumption that
the MP is embedded in an inviscid fluid with constant pressure pout, the first boundary
condition is given by
The model is composed of two phases with individual momentum balances. Hence, we
need a second boundary condition to close the model equations. Neglecting permeation of gel
or fluid through the boundary, the second boundary condition is
Free boundary conditions require solving the momentum balances at the boundary, whose
position must be determined in the course of solving the evolution equations. To circumvent
this problem, we formulate the model in a co-moving body reference frame. The details of the
transformation from the laboratory to the body reference frame can be found in [37?39]. S1
Fig displays a visual comparison of a quantity plotted in the body reference and the laboratory
The stress of the passive fluid with viscosity ?f is given by ?f = ?f@xv. The gel?s stress can be
decomposed in a passive part, ?ve, and an active part, ?act. The passive part is described as a
viscoelastic Kelvin-Voigt solid with sve ? E@xu ? Zg@xu_ [50, 57] where E denotes Young?s
modulus and ?g the dynamic viscosity. For a discussion on the effects of different linear and
nonlinear viscoelastic models see [36, 58]. The active stress is assumed to be governed by a
chemical regulator c, which is usually identified as calcium [13, 59, 60], according to
In this formulation, the homogeneous part of the active stress T0 is inhibited by calcium
concentration c with a coupling strength ? > 0.
In , we identified a heterogeneous substrate friction as a requisite for motion of the
MP?s center of mass (COM). While the exact nature of the Physarum friction dynamics is not
known, recent experiments indicate that MP exhibit nonlinear slip-stick friction with an
underlying substrate resulting in a heterogeneous friction . We assume that there are
adhesive bonds between gel and substrate which break once the force acting on them is larger than
a critical value and that this force depends on the local gel speed ju_ j . Therefore, if ju_ j is
larger than the critical slip speed vslip these bonds will break and the friction coefficient will
decrease locally, yielding
g?ju_ j? ? g0: ?1
with 0 < ? < 1. With ? = 0 the friction coefficient is always homogeneous, higher values allow
for heterogeneous friction coefficient values. In the following, we call ? the slip-ratio and ?0
the base friction coefficient.
The fluid phase contains dissolved chemical species that can perform regulatory activities.
Following , we consider an advection-diffusion-reaction dynamics for two control species
with local instantaneous concentrations: calcium c and a control species a that chemically
interacts with calcium. Calcium determines the gel?s active tension while the control species
represents all biochemical interactions between calcium and other components of the cytosol.
We assume an oscillatory Brusselator-type kinetics for a and c .
Both species are dissolved in the fluid and advected with its flow. Furthermore, they diffuse
with coefficient Dc and Da, respectively. We assume that both species cannot cross the
boundary, resulting in no-flux boundary conditions.
Linearizing with respect to the gel strains @xu yields advection-diffusion-reaction equations
with the relative velocity of the fluid to the gel v u_ as the advection velocity. In the body
reference frame the equations for c and a read
Here, ? = 0.105/s represents the temporal scale of the calcium kinetics which can be used to
fit the experimentally observed time scale of the mechano-chemical waves.
Fig 1. Sketch of the model. Calcium regulates the gel?s active tension and varying calcium concentration levels cause gel deformations. When the gel
moves, there is nonlinear friction between gel and substrate. Moreover, pressure gradients arise and there is friction between both phases. Both cause
the fluid to start flowing and advecting the dissolved chemical species.
The homogeneous steady state (HSS) of the model is given by c = A and a = B/A. The HSS
destabilizes through a Hopf-Bifurcation if B > Bcr = 1 + A2 . Here, we fix A = 0.8 and then
vary B above its critical value Bcr. With the inclusion of a calcium kinetics, the total amount of
calcium is not conserved anymore. In addition, the calcium kinetics can introduce an axis of
polarity [39, 40]. The case with calcium conservation was examined in earlier studies for
resting  and moving poroelastic droplets .
Fig 1 gives an overview of the different elements in the model. Calcium regulates the gel?s
activity. Spatial variations in the calcium concentration yield a heterogeneous active tension
which results in deformation and motion of the gel. Once the gel moves, there is nonlinear
friction between gel and substrate. Furthermore, the gel?s motion induces pressure gradients.
Due to these gradients and friction between the gel and the fluid phase, the fluid starts to flow
and the dissolved chemical species are advected with the fluid, closing the feedback loop
between the different elements in the model.
In summary, the model equations are given by
Following , we introduce the Pe?clet number Pe = ?/(Dc?) as a measure for the ratio of
diffusive to advective time scales to characterize the strength of the active tension . We also
define the dimension-less ratio of the Pe?clet number to the critical Pe?clet number for the onset
of the mechano-chemical instability without a calcium kinetics  as F = ?A/(Dc?(1 + A)2).
For F > 1(F < 1), the HSS is stable (unstable).
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
u ? u_ ? v ? 0; c ? c0 ? A; a ? a0 ? B=A:
The length L remains constant due to the incompressibility of the medium. We use the
ter of mass (COM) position xCOM ? 0L u?x?dx as a measurement of the position. Note that we
will compare the COM?s motion to our results from  where we utilized the location of the
left boundary as the position and the COM always remained fixed. In addition, we utilize the
temporal distance between subsequent peaks in the calcium concentration c as a measure for
the period P of the internal dynamics.
Depending on the parameter values of active tension (F), nonlinear friction (?, vslip, ?0) and
calcium kinetics (B, ?), we can observe resting, stationary MPs and MPs performing three
types of oscillatory motion: first, with fixed COM; second, with back and forth moving COM;
third, with back and forth moving COM together with net displacement of the COM (net
motion). In addition, we analyze the effect of parameters that are accessible in the experiment
like the length L and the base friction coefficient ?0.
Nonlinear slip-stick friction allows for COM motion
First, we analyze the effect of the nonlinear substrate friction without calcium kinetics (? = 0),
i. e. the calcium dynamics is governed by a pure advection-diffusion dynamics and the total
amount of calcium is conserved. Afterwards, we introduce a nonlinear calcium kinetics to
establish an axis of polarity.
We vary the slip-ratio ? and the slip-velocity vslip. For spatially uniform substrate friction
(? = 0), there are gel deformations, and the boundaries change their position, but the COM?s
position remains fixed as described in . Increasing ? > 0 gives rise to spatially
heterogeneous friction allowing for COM motion. For small increases of ?, the mode of motion usually
remains the same as with a homogeneous friction, but it is now accompanied by COM motion
of the same type but with a smaller magnitude. In general, the COM motion?s amplitude will
increase when increasing ? (S5 Fig). However, large values of ? might result in a change of the
mode of motion, for example, an initially periodic dynamics might become irregular. For
values of vslip > max ?ju_ j? the friction is always in the sticking regime. Therefore, the friction
coefficient is homogeneous. Likewise, for low values of vslip the moving MP is almost always
in the regime of slipping friction resulting in an effectively homogeneous friction coefficient of
? = ?0(1 ? ?). Intermediate values of vslip result in a heterogeneous distribution of the friction
Comparable to our results in , the position over time remains fixed or undergoes
periodic (irregular) motion of the COM and the boundaries together with a periodic (irregular)
calcium dynamics, as shown in Fig 2. However, we found no cases of net motion. As discussed
in , the lack of net motion gives rise to the question of why the time-averaged position
vanishes for all of these cases. While the calcium distributions produced by pure
advection-diffusion dynamics may look asymmetric at certain instances in time, the long time-averaged
Fig 2. COM (orange) and boundary (blue) trajectories (top row) and calcium dynamics (bottom row) for regular (left column) and irregular
(right column) motion with nonlinear substrate friction without calcium kinetics (? = 0). The nonlinear substrate friction creates a spatially
heterogeneous friction which allows for COM motion (orange line) together with motion of the boundaries (blue). The regular calcium dynamics
(bottom left) results in oscillatory back and forth motion of COM as well as the boundaries (top left). An irregular calcium dynamics (bottom right)
creates irregular motion (top right). There is no net motion for both cases. S1 and S2 Figs display a visual comparison of the calcium concentration
plotted in the body reference and the laboratory frame. Parameters: vslip = 6.0 ?m/s ? = 0.25, ?0 = 10?5 kg/s, L = 130 ?m, Pe = 50 (left) and vslip =
30.0 ?m/s, ? = 0.3, ?0 = 10?5 kg/s, Pe = 9.5, ? = 10?4 kg ?m?3 s?1, E = 0.01 kg ?m?1 s?2, ?g = 0.01 kg ?m?1 s?1, ?f = 2 ? 10?8 kg ?m?1 s?1 (right).
distribution is always symmetric. This indicates that the front-back symmetry is not broken on
P, and that an additional mechanism to establish an axis of polarity is
Note that the exact nature of Physarum?s friction dynamics can not be inferred from the
cium concentration c, the local contraction amplitude @xu (which is proportional to the height
h ), and more complex slip-stick models led to qualitatively equivalent results for the onset
Nonlinear calcium kinetics creates an axis of polarity
One self-organized way that leads to the emergence of uni-directional traveling
mechanochemical waves which establishes an axis of polarity that is stable on timescales of t P is to
introduce a nonlinear calcium kinetics as studied in [39, 40]. Introducing this calcium kinetics
allows for a temporal variation of the total amount of calcium. The amplitude of calcium
waves can vary while traveling waves can annihilate on collision with a boundary. This is in
contrast to the behavior with a pure advection-diffusion dynamics where calcium is conserved
and waves always get reflected at the boundaries.
The linear stability of the HSS (Eq 7) against small perturbations is analyzed in detail in [39,
40]. In  five regions with qualitatively different dispersion relations are given: i) the HSS is
stable for small F and B. ii) Upon increasing the chemical activity B > Bcr the HSS undergoes a
supercritical Hopf bifurcation. iii) Decreasing B < Bcr and increasing the mechanical activity
F > 1 gives rise to an oscillatory short-wavelength instability as already identified and
discussed in . iv) For B > Bcr and F > 1 both instabilities are present and more complex wave
patterns can emerge. v) A further increase of F results in the real eigenvalues maximum to
become purely real for small k. However, for larger k these eigenvalues still have an imaginary
component 6? 0. Note that we analyzed the linear stability of the model with linear friction
(fsub ? g0u_ ), as the nonlinear friction terms vanishes in the linear approximation. The forms
of the typical resulting dispersion relations are analogous to the ones displayed in .
In numerical simulations, we find modes defining an axis of polarity that is stable on
timescales of t P for parameters in region iv. These modes are distinguished by a spatially
asymmetric long time-averaged calcium distribution. This is in contrast to the averaged calcium
distributions produced by the pure advection-diffusion dynamics which are always spatially
symmetric. Such an axis of polarity that remains stable on long timescales will be referred to as
stable polarity in the remainder of this work.
Nonlinear calcium kinetics combined with nonlinear substrate friction
result in net motion
We identified the two ingredients we need to build a model with net motion. For appropriate
values for the parameters of the calcium kinetics and the substrate friction oscillatory back and
forth motion with net motion of the COM is observed. In the following, we characterize two
types of oscillations with net motion occurring in different parameter regions.
In the first type (type 1), calcium waves emerge periodically at the front traveling backwards
with decreasing amplitude together with backward motion of both boundaries. While traveling
backward, the calcium wave causes forward motion of gel and COM. Upon collision with the
rear boundary, the calcium concentration first peaks and the wave is then annihilated. Once
the calcium wave has decayed, the tension in the material relaxes and there is no motion until
the next calcium wave arises. The slip velocity vslip is only exceeded while a wave is
propagating. Therefore, there is only COM motion during wave propagation, and the COM remains at
a constant position otherwise.
The continuous emergence of calcium waves at the front and their annihilation at the back
together with the asymmetric time-averaged calcium distribution indicate the establishment of
a stable polarity. This dynamics results in oscillatory back and forth motion of COM and
boundaries together with net motion.
The period of this motion is P 95 s. In each period the COM moves forward by 18 ?m
which results in a net speed of 0.2 ?m/s. Fig 3 shows gel velocity, calcium dynamics as well as
boundary (blue) and COM (orange) position for the first type of oscillations with net motion.
This type is the most commonly observed case of COM oscillations with net motion, and we
observe it for a wide range of parameters.
For the second type (type 2) of COM oscillations with net motion, calcium waves emerge
alternating from both sides traveling towards the opposite boundary. Upon approaching the
front (rear), the calcium waves coincide with forward (backward) motion of the boundaries.
When the wave hits the boundary, the calcium concentration peaks, sparking the emergence
of a new wave traveling into the opposite direction. Waves emerging from the front have a
Fig 3. Gel velocity (top), calcium dynamics (middle) and trajectories (bottom) of boundary (blue) and COM (orange) for type 1 motion.
Backward traveling calcium waves emerge periodically at the front, together with backward motion of the boundaries. Once the calcium wave is
traveling backward, it causes forward gel flow and COM motion. On their arrival at the rear, the calcium concentration peaks, but the wave is
annihilated upon collision with the boundary. The resulting forward gel motion is of a greater amplitude than the backward motion resulting in net
motion with vnet = 0.2 ?m/s. S3 Fig displays a visual comparison of the calcium concentration plotted in the body reference and the laboratory frame.
Parameters B = 3.5, ?0 = 7 ? 10?6 kg/s, vslip = 2.5 ?m/s, ? = 0.15, L = 160 ?m and F = 12.3.
higher amplitude than waves from the back as displayed in S6 Fig. The higher amplitude of
waves traveling from front to rear indicate a stable polarity and results in net motion.
The boundaries as well as the COM oscillate with the same period of P
amplitude of the COM motion is smaller than the amplitude of the boundaries motion. The
COM advances the boundaries by a quarter period. In each cycle, the COM moves forward by
COM and boundary trajectories for type 2 motion. The gel velocity resembles the pattern
found in , but now the calcium kinetics creates a stable polarity. This type only appears for
a few specific parameter combinations.
In addition to the two types of oscillations with net motion described above, we find several
modes with time-periodic or irregular switching polarity (S7 Fig).
Model reproduces experimentally observed types of movement
Comparing our simulations with Physarum experiments reveals several qualitative and
quantitative similarities. In experiments, MPs explore their surrounding with an oscillatory forward
and backward motion [21, 24]. Their forward motion is of larger magnitude than their
Fig 4. Gel velocity (left), calcium dynamics (middle) and trajectories (bottom) of boundary (blue) and COM (orange) for type 2 motion. Calcium
waves emerge alternating from front and back traveling towards the opposite side. The collision with a boundary sparks a new wave traveling
backwards. Waves emerging at the front have a larger amplitude than waves originating at the rear resulting in net motion. S4 Fig displays a visual
comparison of the calcium concentration plotted in the body reference and the laboratory frame. Parameters: B = 2, ?0 = 10?5 kg/s, vslip = 1.54 ?m/s, ? =
0.1, L = 125 ?m and F = 18.5.
backward motion, resulting in net motion in each cycle. An experimental MP trajectory is
shown in Fig 6b in .
Although experimental measurements of the free calcium dynamics are noisy, forward
traveling calcium waves can be discerned that annihilate on collision with the front .
Moreover, the velocities of two different quantities can be measured: endo- and ectoplasm . The
ectoplasm is gel-like and connected to the MP?s membrane while the endoplasm is fluid-like.
Strength and direction of the ectoplasm?s velocity correlate with measured traction forces
exerted on the substrate. In the model, the gel is viscoelastic and exhibits friction with the
substrate. Hence, the velocities of gel and ectoplasm should be compared. For the more common
peristaltic type of MP motion, the ectoplasm?s velocity alternates periodically between forward
and backward direction at a fixed position. While the velocity?s direction at front and back are
equal, the center part is moving into the opposite direction with a weaker amplitude than at
the boundaries. Forward ectoplasm motion has a larger amplitude than backward motion,
resulting in a net speed of vexp 0.05-0.25 ?m/s with an internal period of Pexp 80 s-130 s
[13, 23]. A space-time plot of the ectoplasm velocity is shown in  (Fig 7a).
Both types of simulated movement exhibit oscillatory COM motion with net motion
in each cycle. For type 1 motion (Fig 3) the period of 95 s and net speed of 0.2 ?m/s are
comparable to the experiment. However, the COM is only moving while a calcium wave is
propagating and is at rest otherwise which has not been observed in the experiment. Type 2
motion (Fig 4) features continuous forward and backward motion with a net speed of
0.21 ?m/s and a period of 14 s. Aside from the substantially shorter temporal period, this type
does more closely resemble the experimental observations.
The calcium dynamics of both types feature traveling waves. Type 1 motion exhibit only
backward traveling waves that annihilate at the rear. However, for type 2 motion waves get
partly reflected and there are forward and backward traveling calcium waves. In the
experiment, only forward traveling waves that annihilate at the front can be observed .
In addition, we can compare the experimental ectoplasm velocity with the simulated gel
velocity. For type 1 motion, each calcium wave causes huge gel deformations that are
accompanied by COM motion. When the calcium wave has abated, the stress in the medium relaxes
and after 25 s all deformations have decayed. There is no further motion until the next
calcium wave emerges. The velocity patterns in this type of motion have not been observed
However, there are several similarities between the velocity patterns in peristaltic MP and
type 2 motion. In experiment as well as simulation the gel?s velocity alternates between
forward and backward direction at a given position. Usually, the gel?s velocity possesses the
same direction at front and back, whereas there is opposing motion at the center. The
forward velocity has a larger amplitude, which results in net motion towards the front.
However, there are also differences: In the experiment, the ectoplasm?s velocity at the rear is
higher than at the front which is not the case in our simulations under the chosen parameter
values. As mentioned above, the period of 14 s is shorter than experimentally measured
In the parameter plane spanned by base friction strength ?0 and length L the type of motion as
well as net and mean speed changes as displayed in Fig 5. The mean speed is measured by the
averaged speed of the boundaries. For very high values of ?0 ( 3 ? 10?5 kg/s) global calcium
oscillations emerge without any gel or COM motion, regardless of the length. For low values of
?0 (< 10?6 kg/s), there is effectively no friction between gel and substrate. In this case, the
equations do not possess full rank and we omit showing these results.
In the intermediate range the length L strongly affects the emerging dynamics. For ?0 =
10?5 kg/s global calcium oscillations occur when L 100 ?m. Increasing L above 110 ?m gives
rise to type 1 motion with a low net speed of 0.03 ?m/s (Fig 5, top). With a further increase of
L the net speed rises up to 0.16 ?m/s for L = 170 ?m. When decreasing the base friction
coefficient to ?0 = 5 ? 10?6 kg/s, the net speed increases to 0.21 ?m/s where it reaches a maximum. A
further decrease yields lower net speeds. At 2 ? 10?6 kg/s a transitions to oscillations of the
COM without net motion is observed.
Additionally, we find a critical length Lcr 120 ?m for the transition between global
calcium oscillations and states with motion of the boundaries (Fig 5, bottom). Our linear stability
analysis predicts that there is a critical length for the transition to an oscillatory
short-wavelength instability (green line). This prediction is close to the transition from global calcium
oscillations to COM oscillations without net motion (bottom panel).
The existence of a critical length as well as the increasing net speed for longer MPs are
qualitatively in line with experimental observations. It was found that there is a critical MP size for
the onset of locomotion which is around 100 ?m-200 ?m [13, 61]. Furthermore, previous
experiments show that an increase in the longitudinal length increases the net speed [61, 62].
Fig 5. Net (top) and mean speed (bottom) for different strengths of base substrate friction ?0 and length L. Larger MPs possess both a higher net
and mean speed. The net speed reaches a maximum for a medium base friction coefficient of ?0 = 6 ? 10?6 kg/s while the mean speed increases with
decreasing friction. There is a critical length of Lcr 120 ?m in the model and L needs to be larger than Lcr to transition from global calcium oscillations
to states with motion of the boundaries. This is in qualitative agreement with the linear stability analysis that predicts the transition to an oscillatory
short-wavelength instability (green line, bottom) at 130 ?m for ?0 = 6 ? 10?6 kg/s. Parameters: B = 3.5, vslip = 2.5 ?m/s, ? = 0.15 and F = 12.3.
Note that MPs start to branch when they grow above 300 ?m-500 ?m [13, 21, 63] and that the
model cannot capture any effects caused by this.
By changing the temporal scale ? of the calcium kinetics (Eq 5) we can adjust the period P
of the internal dynamics for type 1 motion. In general, higher (lower) values of ? result in a
shorter (longer) period P. The net speed decreases when increasing the period P (Fig 6). With
a decreasing period the intervals between two calcium waves become shorter. As each wave
results in net motion, the net speed decreases with higher periods until there is a transition to
global calcium oscillations without any motion for periods of more than P 145 s. In the
experiment, the maximum period observed for moving MP is 128 s, which is qualitatively in
line with the maximum period of 145 s for moving MP. Consistent with our results above, a
larger slip-ratio ? increases the net speed.
To our knowledge, there is no published experimental data where the MP net speed is
compared to the period of its internal dynamics. Our modeling results may hence be tested in
future more detailed experimental studies of motion of MP.
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
Fig 6. A higher period P of the internal dynamics results in lower net speeds for type 1 motion. For a period of more than P > 145 s there is a
transition to global calcium oscillations without any motion. The experimentally observed periods are between 78 s and 127 s, taken from .
Parameters: B = 3.5, vslip = 3.0 ?m/s and F = 12.3.
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 .
With this extended model, we explore the conditions for self-organized motion of
Physarum MP. We identified varying modes of locomotion ranging from global calcium
oscillations without motion to spatio-temporal oscillations with and without net motion. Our study
was restricted to one spatial dimension which can be rationalized by the experimental evidence
that moving MPs have an elongated shape and motion is mostly caused by deformation waves
along the longitudinal axis .
With the extended model, we could identify two different types of COM oscillations with
net motion in addition to modes with time-periodic or irregular switching polarity. The more
frequent type is characterized by mechano-chemical waves traveling from the front towards
the rear that are accompanied by COM motion (Fig 3). The second, less frequent type is
characterized by mechano-chemical waves that appear alternating from front and back (Fig 4).
While both types exhibit oscillatory forward and backward motion with net motion in each
cycle, in particular the trajectory and gel flow pattern of the second type resemble experimental
measurements of peristaltic MP motion . Interestingly, there are also experimental
observations of two different types of oscillations with net motion in addition to disorganized
Further, we varied parameters that are accessible in experiments, such as the period of the
internal dynamics P, the length L, and the base substrate friction coefficient ?0.
In the parameter plane spanned by the base friction coefficient (?0) and the length (L), the
model predicts that MPs need to be longer than a critical length of 120 ?m to transition
from global calcium oscillations to states with motion of the boundaries (Fig 5). This result
agrees with the linear stability analysis which indicated a transition to an oscillatory
shortwavelength for 125 ?m-130 ?m. Furthermore, experimental studies found a minimal length
of 100 ?m-200 ?m for MPs to start their motion [13, 61]. In addition, the net speed in the
simulations increases with length L which was also observed in experiments .
The net speed becomes maximal for intermediate values of ?0 and larger slip-ratios ? result
in a larger net speed (Fig 6). While the exact nature of Physarum?s friction dynamics can not
be inferred from the available experimental data, introducing a nonlinear friction dynamics is
crucial for COM motion. Future experiments may provide more quantitative information on
the interactions between substrate and MP which could be used to improve the model.
The net speed in the simulations decreases monotonically with an increasing period (Fig 6).
For parameters leading to a period P > 145 s, we only found global calcium oscillations
without any motion. This is qualitatively in line with experimental results, where the maximum
observed period is 128s for moving MP .
With the results from this work, we have been able to identify components that are essential
for different types of MP locomotion. It is known from  that a feedback loop between a
passive chemical regulator, active mechanical deformations, and induced flow is sufficient for
the formation of spatio-temporal contraction patterns. In , we extended the model by
employing free boundary conditions and linear substrate friction which yielded oscillatory and
irregular motion of the MP?s boundaries with a resting COM. In the present work, we
introduced nonlinear stick-slip substrate friction that enables COM motion. Moreover, inclusion of
a nonlinear oscillatory calcium kinetics leads to a self-organized stable polarity which
eventually allows for net motion. Fig 7 summarizes the connection between these components and
describes the corresponding types of locomotion.
Fig 7. Overview of the different levels in the model and corresponding types of locomotion.
Much larger forms of Physarum like mesoplasmodia and extended networks show a clear
separation of gel and fluid-like phases [8, 12, 63, 64]. Recently, a study by Weber et. al 
identified that differences in the activity between two phases can lead to a phase separation.
Future extensions of our model may include this aspect by treating the phase composition as a
spatially dependent local variable that includes transitions between both phases.
S1 Fig. Regular motion with nonlinear substrate friction without calcium kinetics in in
body reference (top) and lab frame (bottom). We solve our model equations in the gel?s
body reference frame and the resulting quantities are defined in this frame. However,
observers are located in the lab frame. The quantity?s transformation from body reference to lab
frame is given by the displacement field u with X0 = x0 + u(x0). Here, x0 the position in the
body reference and X0 is the position in the lab frame. Parameters from Fig 2 (left) in the main
S2 Fig. Irregular motion with nonlinear substrate friction without calcium kinetics in in
body reference (top) and lab frame (bottom). Parameters from Fig 2 (right) in the main text.
We thank Markus Radszuweit for helpful discussions about his previous work on the model
and Shun Zhang and Juan Carlos del A? lamo for discussions about their experimental work.
Conceptualization: Dirk Alexander Kulawiak, Jakob Lo?ber, Markus Ba?r, Harald Engel.
Data curation: Dirk Alexander Kulawiak, Jakob Lo?ber.
Funding acquisition: Markus Ba?r, Harald Engel.
Investigation: Dirk Alexander Kulawiak, Jakob Lo?ber.
Software: Dirk Alexander Kulawiak, Jakob Lo?ber.
Supervision: Markus Ba?r, Harald Engel.
Visualization: Dirk Alexander Kulawiak, Jakob Lo?ber.
Methodology: Dirk Alexander Kulawiak, Jakob Lo?ber, Markus Ba?r, Harald Engel.
Writing ? original draft: Dirk Alexander Kulawiak, Markus Ba?r, Harald Engel.
1. Gross P , Kumar KV , Grill SW . How Active Mechanics and Regulatory Biochemistry Combine to Form Patterns in Development . Annual Review of Biophysics. 2017 ; 46 ( 1 ): 337 - 356 . https://doi.org/10.1146/ annurev- biophys-070816-033602 PMID: 28532214
2. Howard J , Grill SW , Bois JS . Turing's next Steps: The Mechanochemical Basis of Morphogenesis . Nature Reviews Molecular Cell Biology . 2011 ; 12 ( 6 ): 392 - 398 . https://doi.org/10.1038/nrm3120 PMID: 21602907
3. Kolomeisky AB , Fisher ME . Molecular Motors: A Theorist's Perspective. Annual Review of Physical Chemistry . 2007 ; 58 ( 1 ): 675 - 695 . https://doi.org/10.1146/annurev. physchem.58.032806.104532 PMID: 17163836
4. Sanchez T , Chen DTN , DeCamp SJ , Heymann M , Dogic Z. Spontaneous Motion in Hierarchically Assembled Active Matter . Nature . 2012 ; 491 ( 7424 ): 431 - 434 . https://doi.org/10.1038/nature11591 PMID: 23135402
5. Nishikawa M , Naganathan SR , Ju?licher F , Grill SW . Controlling Contractile Instabilities in the Actomyosin Cortex . eLife. 2017 ; p. e19595. https://doi.org/10.7554/eLife.19595 PMID: 28117665
6. Mayer M , Depken M , Bois JS , Ju?licher F , Grill SW . Anisotropies in Cortical Tension Reveal the Physical Basis of Polarizing Cortical Flows . Nature . 2010 ; 467 ( 7315 ): 617 - 621 . https://doi.org/10.1038/ nature09376 PMID: 20852613
7. Kumar KV , Bois JS , Ju?licher F , Grill SW . Pulsatory Patterns in Active Fluids . Physical Review Letters . 2014 ; 112 ( 20 ):208101. https://doi.org/10.1103/PhysRevLett.112.208101
8. Oettmeier C , Brix K , Doebereiner HG . Physarum Polycephalum-a New Take on a Classic Model System . Journal of Physics D: Applied Physics. 2017 . https://doi.org/10.1088/ 1361 - 6463 /aa8699
9. Werthmann B , Marwan W. Developmental Switching in Physarum Polycephalum: Petri Net Analysis of Single Cell Trajectories of Gene Expression Indicates Responsiveness and Genetic Plasticity of the Waddington Quasipotential Landscape . Journal of Physics D: Applied Physics. 2017; 50 ( 46 ):464003. https://doi.org/10.1088/ 1361 - 6463 /aa8e2b
10. Boisseau RP , Vogel D , Dussutour A. Habituation in Non-Neural Organisms: Evidence from Slime Moulds . Proceedings of the Royal Society B: Biological Sciences . 2016 ; 283 ( 1829 ): 20160446 . https:// doi.org/10.1098/rspb.2016.0446 PMID: 27122563
11. Beekman M , Latty T. Brainless but Multi-Headed: Decision Making by the Acellular Slime Mould Physarum Polycephalum . Journal of Molecular Biology . 2015 ; 427 ( 23 ): 3734 - 3743 . https://doi.org/10.1016/ j.jmb. 2015 . 07.007 PMID: 26189159
12. Oettmeier C , Do?bereiner HG. A Close Look at Amoeboid Locomotion: An Integrated Picture of a Migrating, Starvation-Induced Foraging Unit of Physarum Polycephalum . ACM ; 2016 .
13. Zhang S , Guy R , Lasheras JC , del A?lamo JC. Self-Organized Mechano-Chemical Dynamics in Amoeboid Locomotion of Physarum Fragments . Journal of Physics D: Applied Physics. 2017.
14. Fessel A , Oettmeier C , Do?bereiner HG. Structuring Precedes Extension in Percolating Physarum Polycephalum Networks . Nano Communication Networks . 2015 ; 6 ( 3 ): 87 - 95 . https://doi.org/10.1016/j. nancom. 2015 .04.001
15. Julien JD , Alim K. Oscillatory Fluid Flow Drives Scaling of Contraction Wave with System Size . Proceedings of the National Academy of Sciences . 2018 ; 115 ( 42 ): 10612 - 10617 . https://doi.org/10.1073/ pnas.1805981115
16. Baumgarten W , Hauser MJB . Functional Organization of the Vascular Network of Physarum Polycephalum . Physical Biology . 2013 ; 10 ( 2 ):026003. https://doi.org/10.1088/ 1478 - 3975 /10/2/026003 PMID: 23406784
17. Ba?uerle FK , Kramar M , Alim K. Spatial Mapping Reveals Multi-Step Pattern of Wound Healing in Physarum Polycephalum . Journal of Physics D: Applied Physics. 2017; 50 ( 43 ):434005. https://doi.org/10. 1088/ 1361 - 6463 /aa8a21
18. Alim K , Andrew N , Pringle A , Brenner MP . Mechanism of Signal Propagation in Physarum Polycephalum . Proceedings of the National Academy of Sciences . 2017 ; 114 ( 20 ): 5136 - 5141 . https://doi.org/10. 1073/pnas.1618114114
19. Dirnberger M , Mehlhorn K. Characterizing Networks Formed by P. Polycephalum. Journal of Physics D: Applied Physics. 2017; 50 ( 22 ):224002. https://doi.org/10.1088/ 1361 - 6463 /aa6e7b
20. Takagi S , Ueda T. Emergence and Transitions of Dynamic Patterns of Thickness Oscillation of the Plasmodium of the True Slime Mold Physarum Polycephalum . Physica D: Nonlinear Phenomena . 2008 ; 237 ( 3 ): 420 - 427 . https://doi.org/10.1016/j.physd. 2007 .09.012
21. Rieu JP , Delanoe-Ayari H , Takagi S , Tanaka Y , Nakagaki T. Periodic Traction in Migrating Large Amoeba of Physarum Polycephalum . Journal of The Royal Society Interface . 2015 ; 12 (106). https://doi. org/10.1098/rsif.2015.0099
22. Rodiek B , Hauser MJB . Migratory Behaviour of Physarum Polycephalum Microplasmodia. The European Physical Journal Special Topics . 2015 ; 224 ( 7 ): 1199 - 1214 . https://doi.org/10.1140/epjst/e2015- 02455 -2
23. Rodiek B , Takagi S , Ueda T , Hauser MJB . Patterns of Cell Thickness Oscillations during Directional Migration of Physarum Polycephalum . European Biophysics Journal . 2015 ; 44 ( 5 ): 349 - 358 . https://doi. org/10.1007/s00249- 015 - 1028 - 7 PMID: 25921614
24. Lewis OL , Zhang S , Guy RD , del Alamo JC. Coordination of Contractility, Adhesion and Flow in Migrating Physarum Amoebae . Journal of The Royal Society Interface . 2015 ; 12 (106). https://doi.org/10. 1098/rsif.2014.1359
25. Tanaka Y , Ito K , Nakagaki T , Kobayashi R. Mechanics of Peristaltic Locomotion and Role of Anchoring . Journal of The Royal Society Interface . 2012 ; 9 ( 67 ): 222 - 233 . https://doi.org/10.1098/rsif.2011.0339
26. Kuroda S , Kunita I , Tanaka Y , Ishiguro A , Kobayashi R , Nakagaki T. Common Mechanics of Mode Switching in Locomotion of Limbless and Legged Animals . Journal of The Royal Society Interface . 2014 ; 11 ( 95 ): 20140205 - 20140205 . https://doi.org/10.1098/rsif.2014.0205
27. Lu X , Ren L , Gao Q , Zhao Y , Wang S , Yang J , et al. Photophobic and Phototropic Movement of a SelfOscillating Gel . Chemical Communications . 2013 ; 49 ( 70 ): 7690 - 7692 . https://doi.org/10.1039/ c3cc44480e PMID: 23884557
28. Ren L , She W , Gao Q , Pan C , Ji C , Epstein IR . Retrograde and Direct Wave Locomotion in a Photosensitive Self-Oscillating Gel . Angewandte Chemie International Edition . 2016 ; 55 ( 46 ): 14301 - 14305 . https://doi.org/10.1002/anie.201608367 PMID: 27735127
29. Ren L , Wang M , Pan C , Gao Q , Liu Y , Epstein IR . Autonomous Reciprocating Migration of an Active Material . Proceedings of the National Academy of Sciences . 2017 ; 114 ( 33 ): 8704 - 8709 . https://doi.org/ 10.1073/pnas.1704094114
30. Epstein IR , Gao Q. Photo-Controlled Waves and Active Locomotion . Chemistry-A European Journal . 2017 ; 23 ( 47 ): 11181 - 11188 . https://doi.org/10.1002/chem.201700725
31. Joanny JF , Prost J. Active Gels as a Description of the Actin-myosin Cytoskeleton . HFSP Journal . 2009 ; 3 ( 2 ): 94 - 104 . https://doi.org/10.2976/1.3054712 PMID: 19794818
32. Ko?pf MH , Pismen LM. Non-Equilibrium Patterns in Polarizable Active Layers . Physica D: Nonlinear Phenomena . 2013 ; 259 : 48 - 54 . https://doi.org/10.1016/j.physd. 2013 .05.009
33. Ju?licher F , Kruse K , Prost JF , Joanny J. Active Behavior of the Cytoskeleton . Physics Reports . 2007 ; 449 ( 1-3 ): 3 - 28 . https://doi.org/10.1016/j.physrep. 2007 .02.018
34. Bois JS , Ju?licher F , Grill SW . Pattern Formation in Active Fluids . Physical Review Letters . 2011 ; 106 ( 2 ):028103. https://doi.org/10.1103/PhysRevLett.106.028103 PMID: 21405254
35. Coussy O. Poromechanics . 2nd ed. Chichester, England; Hoboken, NJ : Wiley; 2004 .
36. Taber L , Shi Y , Yang L , Bayly P. A Poroelastic Model for Cell Crawling Including Mechanical Coupling between Cytoskeletal Contraction and Actin Polymerization . Journal of mechanics of materials and structures . 2011 ; 6 ( 1 ): 569 - 589 . https://doi.org/10.2140/jomms. 2011 .6. 569 PMID: 21765817
37. Radszuweit M , Alonso S , Engel H , Ba?r M. Intracellular Mechanochemical Waves in an Active Poroelastic Model . Physical Review Letters . 2013 ; 110 ( 13 ):138102. https://doi.org/10.1103/PhysRevLett.110. 138102 PMID: 23581377
38. Kulawiak DA , Lo?ber J , Ba?r M, Engel H. Oscillatory Motion of a Droplet in an Active Poroelastic TwoPhase Model . Journal of Physics D: Applied Physics. 2019; 52 ( 1 ):014004. https://doi.org/10.1088/ 1361 - 6463 /aae41d
39. Radszuweit M , Engel H , Ba?r M. An Active Poroelastic Model for Mechanochemical Patterns in Protoplasmic Droplets of Physarum Polycephalum . PLoS ONE . 2014 ; 9 ( 6 ): 1 - 12 . https://doi.org/10.1371/ journal.pone. 0099220
40. Alonso S , Strachauer U , Radszuweit M , Ba?r M , Hauser MJ . Oscillations and Uniaxial Mechanochemical Waves in a Model of an Active Poroelastic Medium: Application to Deformation Patterns in Protoplasmic Droplets of Physarum Polycephalum . Physica D: Nonlinear Phenomena . 2016 ; 318 : 58 - 69 . https://doi. org/10.1016/j.physd. 2015 .09.017
41. Urbakh M , Klafter J , Gourdon D , Israelachvili J. The Nonlinear Nature of Friction . Nature . 2004 ; 430 ( 6999 ):525. https://doi.org/10.1038/nature02750 PMID: 15282597
42. Lo?ber J , Ziebert F , Aranson IS . Modeling Crawling Cell Movement on Soft Engineered Substrates. Soft Matter . 2014 ; 10 ( 9 ): 1365 - 1373 . https://doi.org/10.1039/c3sm51597d PMID: 24651116
43. Gardel ML , Sabass B , Ji L , Danuser G , Schwarz US , Waterman CM . Traction Stress in Focal Adhesions Correlates Biphasically with Actin Retrograde Flow Speed . The Journal of Cell Biology . 2008 ; 183 ( 6 ): 999 - 1005 . https://doi.org/10.1083/jcb.200810060 PMID: 19075110
44. Barnhart EL , Allen GM , Ju?licher F , Theriot JA . Bipedal Locomotion in Crawling Cells . Biophysical Journal . 2010 ; 98 ( 6 ): 933 - 942 . https://doi.org/10.1016/j.bpj. 2009 . 10.058 PMID: 20303850
45. Monde?sert-Deveraux S , Allena R , Aubry D . A Coupled Friction-Poroelasticity Model of Chimneying Shows That Confined Cells Can Mechanically Migrate Without Adhesions. Molecular & Cellular Biomechanics . 2018 ; 15 ( 3 ): 22 .
46. Lai JH , del Alamo JC , Rodriguez-Rodriguez J , Lasheras JC . The Mechanics of the Adhesive Locomotion of Terrestrial Gastropods . Journal of Experimental Biology . 2010 ; 213 ( 22 ): 3920 - 3933 . https://doi. org/10.1242/jeb.046706 PMID: 21037072
47. DeSimone A , Tatone A. Crawling Motility through the Analysis of Model Locomotors : Two Case Studies. The European Physical Journal E . 2012 ; 35 ( 9 ). https://doi.org/10.1140/epje/i2012- 12085 -x
48. Radszuweit M. An Active Poroelastic Model for Cytoplasm and Pattern Formation in Protoplasmic Droplets of Physarum Polycephalum . Technische Universita?t Berlin, Fakulta?t II-Mathematik und Naturwissenschaften . Berlin; 2013 .
49. Radszuweit M , Engel H , Ba?r M. A Model for Oscillations and Pattern Formation in Protoplasmic Droplets of Physarum Polycephalum . The European Physical Journal Special Topics . 2010 ; 191 ( 1 ): 159 - 172 . https://doi.org/10.1140/epjst/e2010- 01348 -2
50. Fessel A , Oettmeier C , Wechsler K , Do?bereiner HG. Indentation Analysis of Active Viscoelastic Microplasmodia of P. Polycephalum . Journal of Physics D: Applied Physics. 2018; 51 ( 2 ):024005. https://doi. org/10.1088/ 1361 - 6463 /aa9d2c
51. Alt W , Dembo M. Cytoplasm Dynamics and Cell Motion: Two-Phase Flow Models . Mathematical Biosciences . 1999 ; 156 ( 1-2 ): 207 - 228 . https://doi.org/10.1016/ S0025-5564(98)10067-6 PMID: 10204394
52. Joanny JF , Ju?licher F , Kruse K , Prost JF . Hydrodynamic Theory for Multi-Component Active Polar Gels . New Journal of Physics . 2007 ; 9 ( 11 ): 422 - 422 . https://doi.org/10.1088/ 1367 - 2630 /9/11/422
53. Dembo M , Harlow F. Cell Motion , Contractile Networks, and the Physics of Interpenetrating Reactive Flow. Biophysical Journal . 1986 ; 50 ( 1 ): 109 - 121 . https://doi.org/10.1016/ S0006-3495(86)83444-0 PMID: 3730497
54. Larripa K , Mogilner A. Transport of a 1D Viscoelastic Actin Myosin Strip of Gel as a Model of a Crawling Cell . Physica A: Statistical Mechanics and its Applications . 2006 ; 372 ( 1 ): 113 - 123 . https://doi.org/10. 1016/j.physa. 2006 . 05.008 PMID: 19079754
55. Nickaeen M , Novak IL , Pulford S , Rumack A , Brandon J , Slepchenko BM , et al. A Free-Boundary Model of a Motile Cell Explains Turning Behavior. PLOS Computational Biology . 2017 ; 13 ( 11 ): 1 - 22 . https://doi.org/10.1371/journal.pcbi. 1005862
56. Ko?pf MH , Pismen LM. A Continuum Model of Epithelial Spreading. Soft Matter . 2013 ; 9 ( 14 ): 3727 - 3734 . https://doi.org/10.1039/c3sm26955h
57. Banks HT , Hu S , Kenz ZR. A Brief Review of Elasticity and Viscoelasticity for Solids . Advances in Applied Mathematics and Mechanics . 2011 ; 3 ( 1 ): 1 - 51 . https://doi.org/10.4208/aamm.10- m1030
58. Alonso S , Radszuweit M , Engel H , Ba?r M. Mechanochemical Pattern Formation in Simple Models of Active Viscoelastic Fluids and Solids . Journal of Physics D: Applied Physics. 2017; 50 :434004. https:// doi.org/10.1088/ 1361 - 6463 /aa8a1d
59. Yoshimoto Y , Kamiya N. ATP-and Calcium-Controlled Contraction in a Saponin Model of Physarum Polycephalum . Cell structure and function . 1984 ; 9 ( 2 ): 135 - 141 . https://doi.org/10.1247/csf.9. 135 PMID: 6498946
60. Yoshimoto Y , Matsumura F , Kamiya N. Simultaneous Oscillations of Ca2+ Efflux and Tension Generation in the Permealized Plasmodial Strand of Physarum . Cytoskeleton. 1981 ; 1 ( 4 ): 433 - 443 .
61. Matsumoto K , Takagi S , Nakagaki T. Locomotive Mechanism of Physarum Plasmodia Based on Spatiotemporal Analysis of Protoplasmic Streaming . Biophysical Journal . 2008 ; 94 ( 7 ): 2492 - 2504 . https://doi. org/10.1529/biophysj. 107.113050 PMID: 18065474
62. Kuroda S , Takagi S , Nakagaki T , Ueda T. Allometry in Physarum Plasmodium during Free Locomotion: Size versus Shape, Speed and Rhythm . Journal of Experimental Biology . 2015 ; 218 ( 23 ): 3729 - 3738 . https://doi.org/10.1242/jeb.124354 PMID: 26449972
63. Oettmeier C , Lee J , Do?bereiner HG. Form Follows Function: Ultrastructure of Different Morphotypes of Physarum Polycephalum . Journal of Physics D: Applied Physics. 2018; 51 ( 13 ):134006. https://doi.org/ 10.1088/ 1361 - 6463 /aab147
64. Oettmeier C , Do?bereiner HG. A Lumped Parameter Model of Endoplasm Flow in Physarum Polycephalum Explains Migration and Polarization-Induced Asymmetry during the Onset of Locomotion . PLOS ONE . 2019 ; 14 ( 4 ) :e0215622 . https://doi.org/10.1371/journal. pone.0215622 PMID: 31013306
65. Weber CA , Rycroft CH , Mahadevan L. Differential Activity-Driven Instabilities in Biphasic Active Matter . Physical Review Letters . 2018 ; 120 ( 24 ):248003. https://doi.org/10.1103/PhysRevLett.120.248003 PMID: 29956948