Active Poroelastic Two-Phase Model for the Motion of Physarum Microplasmodia

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.

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title: Self-organized mechano-chemical dynamics in amoeboid locomotion of Physarum fragments.
abstract: The aim of this work is to quantify the spatio-temporal dynamics of flow-driven amoeboid locomotion in small (~100 µm) fragments of the true slime mold Physarum polycephalum. In this model organism, cellular contraction drives intracellular flows, and these flows transport the chemical signals that regulate contraction in the first place. As a consequence of these non-linear interactions, a diversity of migratory behaviors can be observed in migrating Physarum fragments. To study these dynamics, we measure the spatio-temporal distributions of the velocities of the endoplasm and ectoplasm of each migrating fragment, the traction stresses it generates on the substratum, and the concentration of free intracellular calcium. Using these unprecedented experimental data, we classify migrating Physarum fragments according to their dynamics, finding that they often exhibit spontaneously coordinated waves of flow, contractility and chemical signaling. We show that Physarum fragments exhibiting symmetric spatio-temporal patterns of endoplasmic flow migrate significantly slower than fragments with asymmetric patterns. In addition, our joint measurements of ectoplasm velocity and traction stress at the substratum suggest that forward motion of the ectoplasm is enabled by a succession of stick-slip transitions, which we conjecture are also organized in the form of waves. Combining our experiments with a simplified convection-diffusion model, we show that the convective transport of calcium ions may be key for establishing and maintaining the spatiotemporal patterns of calcium concentration that regulate the generation of contractile forces.

title: Coordination of contractility, adhesion and flow in migrating Physarum amoebae
abstract: This work examines the relationship between spatio-temporal coordination of intracellular flow and traction stress and the speed of amoeboid locomotion of microplasmodia of Physarum polycephalum. We simultaneously perform particle image velocimetry and traction stress microscopy to measure the velocity of cytoplasmic flow and the stresses applied to the substrate by migrating Physarum microamoebae. In parallel, we develop a mathematical model of a motile cell which includes forces from the viscous cytosol, a poro-elastic, contractile cytoskeleton and adhesive interactions with the substrate. Our experiments show that flow and traction stress exhibit back-to-front-directed waves with a distinct phase difference. The model demonstrates that the direction and speed of locomotion are determined by this coordination between contraction, flow and adhesion. Using the model, we identify forms of coordination that generate model predictions consistent with experiments. We demonstrate that this coordination produces near optimal migration speed and is insensitive to heterogeneity in substrate adhesiveness. While it is generally thought that amoeboid motility is robust to changes in extracellular geometry and the nature of extracellular adhesion, our results demonstrate that coordination of adhesive forces is essential to producing robust migration.

title: Mechanics of peristaltic locomotion and role of anchoring
abstract: Limbless crawling is a fundamental form of biological locomotion adopted by a wide variety of species, including the amoeba, earthworm and snake. An interesting question from a biomechanics perspective is how limbless crawlers control their flexible bodies in order to realize directional migration. In this paper, we discuss the simple but instructive problem of peristalsis-like locomotion driven by elongation–contraction waves that propagate along the body axis, a process frequently observed in slender species such as the earthworm. We show that the basic equation describing this type of locomotion is a linear, one-dimensional diffusion equation with a time–space-dependent diffusion coefficient and a source term, both of which express the biological action that drives the locomotion. A perturbation analysis of the equation reveals that adequate control of friction with the substrate on which locomotion occurs is indispensable in order to translate the internal motion (propagation of the elongation–contraction wave) into directional migration. Both the locomotion speed and its direction (relative to the wave propagation) can be changed by the control of friction. The biological relevance of this mechanism is discussed.

title: Common mechanics of mode switching in locomotion of limbless and legged animals
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title: Photophobic and phototropic movement of a self-oscillating gel.
abstract: A photosensitive self-oscillating gel that incorporates the Belousov-Zhabotinsky reaction can undergo rhythmic mechanical oscillations. We exploit the dependence of the oscillation frequency on light intensity to generate both photophobic and phototropic movement of the gel under differential illumination. Our findings may be used in designing intelligent sensors that can execute biomimetic behaviours.

title: Retrograde and Direct Wave Locomotion in a Photosensitive Self-Oscillating Gel.
abstract: Crawling motion mediated by retrograde and direct waves, that is, in the opposite or the same direction, respectively, as the muscular wave that generates it, is a fundamental mode of biological locomotion, from which more complex and sophisticated locomotion modes involving outgrowths such as limbs and wings may have evolved. A detailed general description of muscular wave locomotion and its relationship with other modes of locomotion is a challenging task. We employ a model of a photosensitive self-oscillating gel, in which chemical pulse waves and a stimulus-responsive medium play roles analogous to nerve pulses and deformable muscles in an animal, to generate retrograde and direct waves under non-uniform illumination. Analysis reveals that the directional locomotion arises from a force asymmetry that results in unequal translation lengths in the push and pull regions associated with a pulse wave. This asymmetry can be modulated by the kinetic parameters of the photosensitive Belousov-Zhabotinsky reaction and the performance parameters of the gel, enabling a transition between retrograde and direct wave locomotion.

title: Autonomous reciprocating migration of an active material
abstract: We analyze the reciprocating locomotion of a photosensitive gel undergoing an oscillating chemical reaction in a temporally constant gradient illumination environment to examine whether periodic migration can arise as an adaptive response of a system’s internal dynamics, analogous to biological functions like growth or breeding, to specific spatial variations in resource distribution, such as food, temperature, or light intensity, caused by geography or seasonal transition. Even in this relatively simple system, kinematic switching in the mode of locomotion under spatially varying illumination levels leads to reciprocating periodic motion, suggesting that such behavior may occur widely in organisms as a response to a variety of spatially distributed environmental stimuli to enhance their survival and reproduction. Periodic to-and-fro migration is a sophisticated mode of locomotion found in many forms of active matter in nature. Providing a general description of periodic migration is challenging, because many details of animal migration remain a mystery. We study periodic migration in a simpler system using a mechanistic model of a photosensitive, active material in which a stimulus-responsive polymer gel is propelled by chemical waves under the regulation of an illumination gradient sensed by the gel, which plays a role analogous to the environment in periodic animal migration. The reciprocating gel migration results from autonomous transitions between retrograde and direct wave locomotion modes arising from the gradient distribution of the illumination intensity. The local dynamics of the chemical waves modulates the asymmetry between push and pull forces to achieve repeated reorientation of the direction of locomotion. Materials that display similar intelligent, self-adaptive locomotion might be tailored for such functions as drug delivery or self-cleaning systems.

title: Photo-Controlled Waves and Active Locomotion.
abstract: Waves of chemical concentration, created by the interaction between reaction and diffusion, occur in a number of chemical systems far from equilibrium. In appropriately chosen polymer gels, these waves generate mechanical forces, which can result in locomotion. When a component of the system is photosensitive, light can be used to modulate and control these waves. In this Concept article, we examine various forms of photo-control of such systems, focusing particularly on the Belousov-Zhabotinsky oscillating chemical reaction. The phenomena we consider include image storage and image processing, feedback-control and feedback-induced clustering of waves, and phototropic and photophobic locomotion. Several of these phenomena have analogues in or potential applications to biological systems.

title: Active gels as a description of the actin‐myosin cytoskeleton
abstract: This short review presents a qualitative introduction to the hydrodynamic theory of active polar gels and its applications to the mechanics of the cytoskeleton. Active polar gels are viscoelastic materials formed by polar filaments maintained in a nonequilibrium state by constant consumption of energy. In the cytoskeleton of eukaryotic cells, actin filaments are treadmilling and form a viscoelastic gel interacting with myosin molecular motors driven by the hydrolysis of adenosine triphosphate; one can thus consider the actomyosin cytoskeleton as an active polar gel. The hydrodynamic description is generic as it only relies on symmetry arguments. We first use the hydrodynamic approach to discuss the spontaneous generation of flow in an active polar film. Then we give two examples of applications to lamellipodium motility and to instabilities of cortical actin.

title: Non-equilibrium patterns in polarizable active layers
abstract: We formulate and explore a generic continuum model of a polarizable active layer with nonlinear elasticity and chemo-mechanical interactions. Homogeneous solutions of the model equations exhibit a monotonic long-wave instability when the medium is activated by expansion, and an oscillatory short-wave instability in the case of compressive activation. Both regimes are investigated analytically and numerically. The long-wave instability initiates a coarsening process, which provides a possible mechanism for the establishment of permanent polarization in spherical geometry.

title: Active behavior of the Cytoskeleton
abstract: This review presents some of our recent results on active polar gels. Active polar gels are viscoelastic materials formed by polar filaments which are maintained in a non-equilibrium state by constant consumption of energy. This non-equilibrium state is characterized by the existence of internal stresses and spontaneous flows. A defining example of an active polar gel is provided by the acto-myosin cytoskeleton of eukaryotic cells. It is formed by actin filaments interacting with myosin molecular motors which are driven by the hydrolysis of adenosine-tri-phosphate (ATP). We first present a hydrodynamic theory of active polar gels. The hydrodynamic equations are generic as they only rely on symmetry arguments. We then use the hydrodynamic approach to study the spontaneous generation of flow in an active polar film and the formation of vortex defects. The last part of this review is devoted to an analysis of the active gel theory in situations which are reminiscent of structures formed by the cytoskeleton in living cells.

title: Pattern Formation in Active Fluids
abstract: We discuss pattern formation in active fluids in which active stress is regulated by diffusing molecular components. Nonhomogeneous active stress profiles create patterns of flow which transport stress regulators by advection. Our work is motivated by the dynamics of the actomyosin cell cortex in which biochemical pathways regulate active stress. We present a mechanism in which a single diffusing species up regulates active stress, resulting in steady flow and concentration patterns. We also discuss general pattern-formation behaviors of reaction-diffusion systems placed in active fluids.

title: A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION.
abstract: Much is known about the biophysical mechanisms involved in cell crawling, but how these processes are coordinated to produce directed motion is not well understood. Here, we propose a new hypothesis whereby local cytoskeletal contraction generates fluid flow through the lamellipodium, with the pressure at the front of the cell facilitating actin polymerization which pushes the leading edge forward. The contraction, in turn, is regulated by stress in the cytoskeleton. To test this hypothesis, finite element models for a crawling cell are presented. These models are based on nonlinear poroelasticity theory, modified to include the effects of active contraction and growth, which are regulated by mechanical feedback laws. Results from the models agree reasonably well with published experimental data for cell speed, actin flow, and cytoskeletal deformation in migrating fish epidermal keratocytes. The models also suggest that oscillations can occur for certain ranges of parameter values.

title: Intracellular mechanochemical waves in an active poroelastic model.
abstract: Many processes in living cells are controlled by biochemical substances regulating active stresses. The cytoplasm is an active material with both viscoelastic and liquid properties. We incorporate the active stress into a two-phase model of the cytoplasm which accounts for the spatiotemporal dynamics of the cytoskeleton and the cytosol. The cytoskeleton is described as a solid matrix that together with the cytosol as an interstitial fluid constitutes a poroelastic material. We find different forms of mechanochemical waves including traveling, standing, and rotating waves by employing linear stability analysis and numerical simulations in one and two spatial dimensions.

title: Oscillatory motion of a droplet in an active poroelastic two-phase model
abstract: We investigate flow-driven amoeboid motility as exhibited by microplasmodia of Physarum polycephalum. A poroelastic two-phase model with rigid boundaries is extended to the case of free boundaries and substrate friction. The cytoskeleton is modeled as an active viscoelastic solid permeated by a fluid phase describing the cytosol. A feedback loop between a chemical regulator, active mechanical deformations, and induced flows gives rise to oscillatory and irregular motion accompanied by spatio-temporal contraction patterns. We cover extended parameter regimes of active tension and substrate friction by numerical simulations in one spatial dimension and reproduce experimentally observed oscillation periods and amplitudes. In line with experiments, the model predicts alternating forward and backward ectoplasmatic flow at the boundaries with reversed flow in the center. However, for all cases of periodic and irregular motion, we observe practically no net motion. A simple theoretical argument shows that directed motion is not possible with a spatially independent substrate friction.

title: 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.

title: Oscillations and uniaxial mechanochemical waves in a model of an active poroelastic medium: Application to deformation patterns in protoplasmic droplets of Physarum polycephalum
abstract: Self-organization in cells often manifests itself in oscillations and waves. Here, we address deformation waves in protoplasmic droplets of the plasmodial slime mould Physarum polycephalum by modelling and experiments. In particular, we extend a one-dimensional model that considered the cell as a poroelastic medium, where active tension caused mechanochemical waves that were regulated by an inhibitor (Radszuweit et al., 2013). Our extension consists of a simple, qualitative chemical reaction–diffusion model (Brusselator) that describes the regulation of the inhibitor by another biochemical species. The biochemical reaction enhances the formation of mechanochemical waves if the reaction rates and input concentrations are near or inside an oscillatory regime. The period of the waves is found to be controlled by the characteristic oscillation period, whereas their wavelength is set by mechanical parameters. The model also allows for a systematic study of the chemical activity at the onset of mechanochemical waves. We also present examples for pattern formation in protoplasmic droplets of Physarum polycephalum including global oscillations where the central region of the droplets is in antiphase to the boundary zone, as well as travelling and standing wave-like uniaxial patterns. Finally, we apply our model to reproduce these experimental results by identifying the active tension inhibitor with the intracellular calcium concentration in the Physarum droplets and by using parameter values from mechanical experiments, respectively knowledge about the properties of calcium oscillations in Physarum. The simulation results are then found to be in good agreement with the experimental observations.

title: The nonlinear nature of friction
abstract: Tribology is the study of adhesion, friction, lubrication and wear of surfaces in relative motion. It remains as important today as it was in ancient times, arising in the fields of physics, chemistry, geology, biology and engineering. The more we learn about tribology the more complex it appears. Nevertheless, recent experiments coupled to theoretical modelling have made great advances in unifying apparently diverse phenomena and revealed many subtle and often non-intuitive aspects of matter in motion, which stem from the nonlinear nature of the problem.

title: Modeling crawling cell movement on soft engineered substrates.
abstract: Self-propelled motion, emerging spontaneously or in response to external cues, is a hallmark of living organisms. Systems of self-propelled synthetic particles are also relevant for multiple applications, from targeted drug delivery to the design of self-healing materials. Self-propulsion relies on the force transfer to the surrounding. While self-propelled swimming in the bulk of liquids is fairly well characterized, many open questions remain in our understanding of self-propelled motion along substrates, such as in the case of crawling cells or related biomimetic objects. How is the force transfer organized and how does it interplay with the deformability of the moving object and the substrate? How do the spatially dependent traction distribution and adhesion dynamics give rise to complex cell behavior? How can we engineer a specific cell response on synthetic compliant substrates? Here we generalize our recently developed model for a crawling cell by incorporating locally resolved traction forces and substrate deformations. The model captures the generic structure of the traction force distribution and faithfully reproduces experimental observations, like the response of a cell on a gradient in substrate elasticity (durotaxis). It also exhibits complex modes of cell movement such as "bipedal" motion. Our work may guide experiments on cell traction force microscopy and substrate-based cell sorting and can be helpful for the design of biomimetic "crawlers" and active and reconfigurable self-healing materials.

title: Traction stress in focal adhesions correlates biphasically with actin retrograde flow speed
abstract: How focal adhesions (FAs) convert retrograde filamentous actin (F-actin) flow into traction stress on the extracellular matrix to drive cell migration is unknown. Using combined traction force and fluorescent speckle microscopy, we observed a robust biphasic relationship between F-actin speed and traction force. F-actin speed is inversely related to traction stress near the cell edge where FAs are formed and F-actin motion is rapid. In contrast, larger FAs where the F-actin speed is low are marked by a direct relationship between F-actin speed and traction stress. We found that the biphasic switch is determined by a threshold F-actin speed of 8–10 nm/s, independent of changes in FA protein density, age, stress magnitude, assembly/disassembly status, or subcellular position induced by pleiotropic perturbations to Rho family guanosine triphosphatase signaling and myosin II activity. Thus, F-actin speed is a fundamental regulator of traction force at FAs during cell migration.

title: Bipedal locomotion in crawling cells.
abstract: Many complex cellular processes from mitosis to cell motility depend on the ability of the cytoskeleton to generate force. Force-generating systems that act on elastic cytoskeletal elements are prone to oscillating instabilities. In this work, we have measured spontaneous shape and movement oscillations in motile fish epithelial keratocytes. In persistently polarized, fan-shaped cells, retraction of the trailing edge on one side of the cell body is out of phase with retraction on the other side, resulting in periodic lateral oscillation of the cell body. We present a physical description of keratocyte oscillation in which periodic retraction of the trailing edge is the result of elastic coupling with the leading edge. Consistent with the predictions of this model, the observed frequency of oscillation correlates with cell speed. In addition, decreasing the strength of adhesion to the substrate reduces the elastic force required for retraction, causing cells to oscillate with higher frequency at relatively lower speeds. These results demonstrate that simple elastic coupling between movement at the front of the cell and movement at the rear can generate large-scale mechanical integration of cell behavior.

title: Indentation analysis of active viscoelastic microplasmodia of P. polycephalum
abstract: Simple organisms like Physarum polycephalum realize complex behavior, such as shortest path optimization or habituation, via mechanochemical processes rather than by a network of neurons. A full understanding of these phenomena requires detailed investigation of the underlying mechanical properties. To date, micromechanical measurements on P. polycephalum are sparse and lack reproducibility. This prompts study of microplasmodia, a reproducible and homogeneous form of P. polycephalum that resembles the plasmodial ectoplasm responsible for mechanical stability and generation of forces. We combine investigation of ultra-structure and dimension of P. polycephalum with the analysis of data obtained by indentation of microplasmodia, employing a novel nonlinear viscoelastic scaling model that accounts for finite dimension of the sample. We identify the multi-modal distribution of parameters such as Young's moduls, Poisson's ratio, and relaxation times associated with viscous processes that cover five orders of magnitude. Results suggest a characterization of microplasmodia as porous, compressible structures that act like elastic solids with high Young's modulus on short time scales, whereas on long time-scales and upon repeated indentation viscous behavior dominates and the effective modulus is significantly decreased. Furthermore, Young's modulus is found to oscillate in phase with shape of microplasmodia, emphasizing that modeling P. polycephalum oscillations as a driven oscillator with constant moduli is not practicable.

title: Cytoplasm dynamics and cell motion: two-phase flow models.
abstract: The motion of amoeboid cells is characterized by cytoplasmic streaming and by membrane protrusions and retractions which occur even in the absence of interactions with a substratum. Cell translocation requires, in addition, a transmission mechanism wherein the power produced by the cytoplasmic engine is applied to the substratum in a highly controlled fashion through specific adhesion proteins. Here we present a simple mechano-chemical model that tries to capture the physical essence of these complex biomolecular processes. Our model is based on the continuum equations for a viscous and reactive two-phase fluid model with moving boundaries, and on force balance equations that average the stochastic interactions between actin polymers and membrane proteins. In this paper we present a new derivation and analysis of these equations based on minimization of a power functional. This derivation also leads to a clear formulation and classification of the kinds of boundary conditions that should be specified at free surfaces and at the sites of interaction of the cell and the substratum. Numerical simulations of a one-dimensional lamella reveal that even this extremely simplified model is capable of producing several typical features of cell motility. These include periodic 'ruffle' formation, protrusion-retraction cycles, centripetal flow and cell-substratum traction forces.

title: Hydrodynamic theory for multi-component active polar gels
abstract: We develop a generic hydrodynamic theory of active fluids with several components. We take into account polar order and consider the case when one component is viscoelastic. Our theory is motivated by the cytoskeleton which is a network of elastic filaments that are coupled to active processes such as the action of motor proteins which can generate relative forces between filaments as they hydrolyze a fuel (ATP). In addition to the filament gel, the system is embedded in a solvent component and free monomers constitute a third component. We derive constitutive material equations for the combined system which include reactive and dissipative couplings as well as the chemical driving by ATP hydrolysis and a possible chiral symmetry of the filaments. As an illustration of these equations, we discuss an active liquid in a simple shear gradient.

title: Cell motion, contractile networks, and the physics of interpenetrating reactive flow.
abstract: In this paper we propose a physical model of contractile biological polymer networks based on the notion of reactive interpenetrating flow. We show how our model leads to a mathematical formulation of the dynamical laws governing the behavior of contractile networks. We also develop estimates of the various parameters that appear in our equations, and we discuss some elementary predictions of the model concerning the general scaling principles that pertain to the motions of contractile networks.

title: Transport of a 1D viscoelastic actin-myosin strip of gel as a model of a crawling cell.
abstract: Cell crawling is an important biological phenomenon because it underlies coordinated cell movement in morphogenesis, cancer and wound healing. This phenomenon is based on protrusion at the cell's leading edge, retraction at the rear, contraction and graded adhesion powered by the dynamics of actin and myosin protein networks. A few one-dimensional models successfully explain an anteroposterior organization of the motile cell, but don't sufficiently explore the viscoelastic nature of the actin-myosin gel. We develop and numerically solve a model of a treadmilling strip of viscoelastic actin-myosin gel. The results show that the strip translocates steadily as a traveling pulse, without changing length, and that protein densities, velocities and stresses become stationary. The simulations closely match the observed forces, movements and protein distributions in the living cell.

title: A free-boundary model of a motile cell explains turning behavior
abstract: To understand shapes and movements of cells undergoing lamellipodial motility, we systematically explore minimal free-boundary models of actin-myosin contractility consisting of the force-balance and myosin transport equations. The models account for isotropic contraction proportional to myosin density, viscous stresses in the actin network, and constant-strength viscous-like adhesion. The contraction generates a spatially graded centripetal actin flow, which in turn reinforces the contraction via myosin redistribution and causes retraction of the lamellipodial boundary. Actin protrusion at the boundary counters the retraction, and the balance of the protrusion and retraction shapes the lamellipodium. The model analysis shows that initiation of motility critically depends on three dimensionless parameter combinations, which represent myosin-dependent contractility, a characteristic viscosity-adhesion length, and a rate of actin protrusion. When the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectories, and the motile behavior is sensitive to conditions at the cell boundary. Scanning of a model parameter space shows that the contractile mechanism of motility supports robust cell turning in conditions where short viscosity-adhesion lengths and fast protrusion cause an accumulation of myosin in a small region at the cell rear, destabilizing the axial symmetry of a moving cell.

title: A continuum model of epithelial spreading
abstract: We propose a continuum model of unconstrained epithelial spreading. The tissue is described as a polarizable and chemo-mechanically interacting layer with neo-Hookean elasticity. Our model reproduces the spontaneous formation of finger-like protrusions commonly observed in experiment. Statistics of the velocity orientation obtained from numerical simulations show a strong alignment within the fingers, as opposed to an isotropic distribution in the bulk. The results faithfully reproduce the relatively faster advance of cells close to the leading edge of the tissue, as well as experimentally observed spatial velocity correlations and stress accumulation within the tissue.

title: Mechanochemical pattern formation in simple models of active viscoelastic fluids and solids
abstract: The cytoskeleton of the organism Physarum polycephalum is a prominent example of a complex active viscoelastic material wherein stresses induce flows along the organism as a result of the action of molecular motors and their regulation by calcium ions. Experiments in Physarum polycephalum have revealed a rich variety of mechanochemical patterns including standing, traveling and rotating waves that arise from instabilities of spatially homogeneous states without gradients in stresses and resulting flows. Herein, we investigate simple models where an active stress induced by molecular motors is coupled to a model describing the passive viscoelastic properties of the cellular material. Specifically, two models for viscoelastic fluids (Maxwell and Jeffrey model) and two models for viscoelastic solids (Kelvin–Voigt and Standard model) are investigated. Our focus is on the analysis of the conditions that cause destabilization of spatially homogeneous states and the related onset of mechano-chemical waves and patterns. We carry out linear stability analyses and numerical simulations in one spatial dimension for different models. In general, sufficiently strong activity leads to waves and patterns. The primary instability is stationary for all active fluids considered, whereas all active solids have an oscillatory primary instability. All instabilities found are of long-wavelength nature reflecting the conservation of the total calcium concentration in the models studied.

title: Simultaneous oscillations of Ca2+ efflux and tension generation in the permealized plasmodial strand of Physarum.
abstract: Calcium is now generally thought to play a key role in regulating a variety of cellular movements. When the plasmodium of Physarum polycephalum was treated with the calcium-ionophore A23187 or the quasi-ionophore amphotericin B, Ca2+ leaked out. Ca2+ efflux into the ambient solution from the plasmodial strand segment was measured by the luminescence of a photoprotein aequorin, and the tensile force production was recorded simultaneously. Ca2+ efflux oscillated with the same period as the cycle of tension generation in the strand, but the phase of cyclic changes in Ca2+ efflux was opposite to that of tension generation. That is, Ca2+ efflux fell in the increasing tension phase and rose in the decreasing tension phase. Cyclic changes in efflux of Ca2+ are provisionally interpreted as reflecting corresponding changes in concentrations of free Ca2+ in the cytoplasm.

title: Periodic traction in migrating large amoeba of Physarum polycephalum
abstract: The slime mould Physarum polycephalum is a giant multinucleated cell exhibiting well-known Ca2+-dependent actomyosin contractions of its vein network driving the so-called cytoplasmic shuttle streaming. Its actomyosin network forms both a filamentous cortical layer and large fibrils. In order to understand the role of each structure in the locomotory activity, we performed birefringence observations and traction force microscopy on excised fragments of Physarum. After several hours, these microplasmodia adopt three main morphologies: flat motile amoeba, chain types with round contractile heads connected by tubes and motile hybrid types. Each type exhibits oscillations with a period of about 1.5 min of cell area, traction forces and fibril activity (retardance) when fibrils are present. The amoeboid types show only peripheral forces while the chain types present a never-reported force pattern with contractile rings far from the cell boundary under the spherical heads. Forces are mostly transmitted where the actomyosin cortical layer anchors to the substratum, but fibrils maintain highly invaginated structures and contribute to forces by increasing the length of the anchorage line. Microplasmodia are motile only when there is an asymmetry in the shape and/or the force distribution.

title: Patterns of cell thickness oscillations during directional migration of Physarum polycephalum
abstract: The functional relationship between the velocity of cell locomotion and intracellular spatial patterns of thickness oscillations in the acellular slime mould Physarum polycephalum was studied. The freely migrating plasmodial cells of 300–800 µm length were tadpole-shaped and displayed thickness oscillations along their longitudinal (body) axis. Two distinct patterns of intracellular thickness oscillations were observed in dependence on the locomotive velocity. The first mode consisted of a single travelling wave that propagated from the rear to the front of the cell. This pattern occurred when the plasmodium migrated slowly. The second mode was a multinodal standing wave that was found during events of fast propagation. Transitions between these two types of cell thickness oscillation patterns took place in narrow propagation velocity intervals. We discuss the possible mechanism leading to these patterns, which are conjectured to modulate both the intracellular pressure and the velocity of free locomotion of the cell.

title: Locomotive mechanism of Physarum plasmodia based on spatiotemporal analysis of protoplasmic streaming.
abstract: We investigate how an amoeba mechanically moves its own center of gravity using the model organism Physarum plasmodium. Time-dependent velocity fields of protoplasmic streaming over the whole plasmodia were measured with a particle image velocimetry program developed for this work. Combining these data with measurements of the simultaneous movements of the plasmodia revealed a simple physical mechanism of locomotion. The shuttle streaming of the protoplasm was not truly symmetric due to the peristalsis-like movements of the plasmodium. This asymmetry meant that the transport capacity of the stream was not equal in both directions, and a net forward displacement of the center of gravity resulted. The generality of this as a mechanism for amoeboid locomotion is discussed.

title: Allometry in Physarum plasmodium during free locomotion: size versus shape, speed and rhythm
abstract: Physarum plasmodium is a giant unicellular organism whose length can vary by more than three orders of magnitude. Using plasmodia ranging in size from 100 μm to 10 cm, we investigated the size dependency of their thickness distributions and locomotion speeds during free locomotion. (1) In the longitudinal direction, the organism is thickest close to the front, and decreases exponentially in thickness towards the rear. The slenderness ratio varies with body size according to a power law, such that large plasmodia are long and flat, whereas small plasmodia are short and thick. (2) The mean locomotion speed is proportional to the mean maximum thickness of the frontal part. By conducting a dimensional analysis, possible physical models are discussed. (3) The intrinsic period of the thickness oscillation, which is related to shuttle streaming (period 1–2 min), increases logarithmically with body size. (4) Various characteristics exhibit size-independent, long-period (20±10 min) oscillations, including speed, shape and intrinsic thickness oscillation period. These variations are closely coupled to formation of the entire cell shape, including undulation of thickness along the longitudinal axis and timing of branching of the frontal tip. Based on these experimental results and those reported previously, we propose a simple mathematical model for cell locomotion.  Allometric studies of Physarum polycepharum reveal the existence of not only scaling law in locomotion speed, cell shape and intrinsic rhythm but also non-scalable rhythm components related to formation of the entire cell shape.

title: Form follows function: ultrastructure of different morphotypes of Physarum polycephalum
abstract: The multinucleate, unicellular slime mold Physarum polycephalum is a highly motile and morphologically diverse giant amoeba. Despite being brainless and lacking neurons, it exhibits 'smart' behavior. There is considerable interest in describing such traits and to investigate the underlying mechanochemical patterns which may hint at universal principles of behavior and decision-making. Furthermore, the slime mold's mechanism of locomotion is unique. It resembles amoeboid movement, but differs from the locomotion of other amoebae in many ways, e.g. in their much larger size and lack of lobopodia. These two aspects, behavior and locomotion, are linked by the cytoskeleton and the overall morphology of P. polycephalum. In this paper, we present a structural analysis of different growth forms (micro-, meso- and macroplasmodia) by transmission electron microscopy (TEM), scanning electron microscopy (SEM), light microscopy, and fluorescence microscopy of F-actin. With these detailed investigations of cellular ultrastructure and morphology, we provide the basis for the analysis of, e.g. viscoelastic and rheological measurements. Our data also provide structural details for the many models that have been constructed for the understanding of locomotion. We conclude that morphological information is vital for the assessment and measurement of material properties.

title: A close look at amoeboid locomotion: An integrated picture of a migrating, starvation-induced foraging unit of Physarum polycephalum
abstract: Physarum polycephalum, cultivated on a glucose-deficient agar surface, forms disconnected foraging units (satellites). Our aim is to shed light onto the amoeboid locomotion of the slime mould, using satellites as reproducible and well-defined models and employing a wide range of techniques. This work was presented at PhysNet 2015.

title: A lumped parameter model of endoplasm flow in Physarum polycephalum explains migration and polarization-induced asymmetry during the onset of locomotion
abstract: The plasmodial slime mold Physarum polycephalum exhibits strong, periodic flow of cytoplasm through the veins of its network. In the special case of mesoplasmodia, a newly described starvation-induced, shape-constant morphotype, this periodic endoplasm streaming is the basis of locomotion. Furthermore, we presume that cytoplasm flow is also involved in signal transmission and signal processing. Mesoplasmodia motility resembles amoeboid locomotion. In contrast to other amoebae, however, mesoplasmodia move without extending pseudopods and retain a coherent, fan-shaped morphology throughout their steady locomotion. Attaining sizes of up to 2 mm2, mesoplasmodia are also much bigger than other amoebae. We characterize this particular type of locomotion and identify patterns of movement. By using the analogy between pulsatile fluid flow through a network of elastic tubes and electrical circuits, we build a lumped model that explains observed fluid flow patterns. Essentially, the mesoplasmodium acts as a low-pass filter, permitting only low-frequency oscillations to propagate from back to front. This frequency selection serves to optimize flow and reduces power dissipation. Furthermore, we introduce a distributed element into the lumped model to explain cell polarization during the onset of chemotaxis: Biochemical cues (internal or external) lead to a local softening of the actin cortex, which in turn causes an increased flow of cytoplasm into that area and, thus, a net forward movement. We conclude that the internal actin-enclosed vein network gives the slime mold a high measure of control over fluid transport, especially by softening or hardening, which in turn leads to polarization and net movement.

title: Differential Activity-Driven Instabilities in Biphasic Active Matter.
abstract: Active stresses can cause instabilities in contractile gels and living tissues. Here we provide a generic hydrodynamic theory that treats these systems as a mixture of two phases of varying activity and different mechanical properties. We find that differential activity between the phases causes a uniform mixture to undergo a demixing instability. We follow the nonlinear evolution of the instability and characterize a phase diagram of the resulting patterns. Our study complements other instability mechanisms in mixtures driven by differential adhesion, differential diffusion, differential growth, and differential motion.