Mechanics of cytogels I: Oscillations in Physarum

Periodicity, mixed-mode oscillations, and multiple timescales in a phosphoinositide-Rho GTPase network

While rhythmic contractile behavior is commonly observed at the cellular cortex, the primary focus has been on excitable or periodic events described by simple activator-delayed inhibitor mechanisms. We show that Rho GTPase activation in nocodazole-treated mitotic cells exhibits both simple oscillations and complex mixed-mode oscillations. Rho oscillations with a 20- to 30-s period are regulated by phosphatidylinositol (3,4,5)-trisphosphate (PIP3) via an activator-delayed inhibitor mechanism, while a slow reaction with period of minutes is regulated by phosphatidylinositol 4-kinase via an activator-substrate depletion mechanism. Conversion from simple to complex oscillations can be induced by modulating PIP3 metabolism or altering membrane contact site protein E-Syt1. PTEN depletion results in a period-doubling intermediate, which, like mixed-mode oscillations, is an intermediate state toward chaos. In sum, this system operates at the edge of chaos. Small changes in phosphoinositide metabolism can confer cells with the flexibility to rapidly enter ordered states with different periodicities.

Mechanobiology in cortical waves and oscillations

Cortical actin waves have emerged as a widely prevalent phenomena and brought pattern formation to many fields of cell biology. Cortical excitabilities, reminiscent of the electric excitability in neurons, are likely fundamental property of the cell cortex. Although they have been mostly considered to be biochemical in nature, accumulating evidence support the role of mechanics in the pattern formation process. Both pattern formation and mechanobiology approach biological phenomena at the collective level, either by looking at the mesoscale dynamical behavior of molecular networks or by using collective physical properties to characterize biological systems. As such they are very different from the traditional reductionist, bottom-up view of biology, which brings new challenges and potential opportunities. In this essay, we aim to provide our perspectives on what the proposed mechanochemical feedbacks are and open questions regarding their role in cortical excitable and oscillatory dynamics.

Rhythmicity and waves in the cortex of single cells

Emergence of dynamic patterns in the form of oscillations and waves on the cortex of single cells is a fascinating and enigmatic phenomenon. Here we outline various theoretical frameworks used to model pattern formation with the goal of reducing complex, heterogeneous patterns into key parameters that are biologically tractable. We also review progress made in recent years on the quantitative and molecular definitions of these terms, which we believe have begun to transform single-cell dynamic patterns from a purely observational and descriptive subject to more mechanistic studies. Specifically, we focus on the nature of local excitable and oscillation events, their spatial couplings leading to propagating waves and the role of active membrane. Instead of arguing for their functional importance, we prefer to consider such patterns as basic properties of dynamic systems. We discuss how knowledge of these patterns could be used to dissect the structure of cellular organization and how the network-centric view could help define cellular functions as transitions between different dynamical states. Last, we speculate on how these patterns could encode temporal and spatial information.

This article is part of the theme issue ‘Self-organization in cell biology’.

Oscillatory fluid flow drives scaling of contraction wave with system size

Flows over remarkably long distances are crucial to the functioning of many organisms, across all kingdoms of life. Coordinated flows are fundamental to power deformations, required for migration or development, or to spread resources and signals. A ubiquitous mechanism to generate flows, particularly prominent in animals and amoebas, is actomyosin cortex-driven mechanical deformations that pump the fluid enclosed by the cortex. However, it is unclear how cortex dynamics can self-organize to give rise to coordinated flows across the largely varying scales of biological systems. Here, we develop a mechanochemical model of actomyosin cortex mechanics coupled to a contraction-triggering, soluble chemical. The chemical itself is advected with the flows generated by the cortex-driven deformations of the tubular-shaped cell. The theoretical model predicts a dynamic instability giving rise to stable patterns of cortex contraction waves and oscillatory flows. Surprisingly, simulated patterns extend beyond the intrinsic length scale of the dynamic instability-scaling with system size instead. Patterns appear randomly but can be robustly generated in a growing system or by flow-generating boundary conditions. We identify oscillatory flows as the key for the scaling of contraction waves with system size. Our work shows the importance of active flows in biophysical models of patterning, not only as a regulating input or an emergent output, but also as a full part of a self-organized machinery. Contractions and fluid flows are observed in all kinds of organisms, so this concept is likely to be relevant for a broad class of systems.

Geometry of Wave Propagation on Active Deformable Surfaces

Fundamental biological and biomimetic processes, from tissue morphogenesis to soft robotics, rely on the propagation of chemical and mechanical surface waves to signal and coordinate active force generation. The complex interplay between surface geometry and contraction wave dynamics remains poorly understood, but it will be essential for the future design of chemically driven soft robots and active materials. Here, we couple prototypical chemical wave and reaction-diffusion models to non-Euclidean shell mechanics to identify and characterize generic features of chemomechanical wave propagation on active deformable surfaces. Our theoretical framework is validated against recent data from contractile wave measurements on ascidian and starfish oocytes, producing good quantitative agreement in both cases. The theory is then applied to illustrate how geometry and preexisting discrete symmetries can be utilized to focus active elastic surface waves. We highlight the practical potential of chemomechanical coupling by demonstrating spontaneous wave-induced locomotion of elastic shells of various geometries. Altogether, our results show how geometry, elasticity, and chemical signaling can be harnessed to construct dynamically adaptable, autonomously moving mechanical surface waveguides.

Nonlinearity in cytoplasm viscosity can generate an essential symmetry breaking in cellular behaviors

The cytoplasms of ameboid cells are nonlinearly viscous. The cell controls this viscosity by modulating the amount, localization and interactions of bio-polymers. Here we investigated how the nonlinearity infers the cellular behaviors and whether nonlinearity-specific behaviors exist. We modeled the developed plasmodium of the slime mold Physarum polycephalum as a network of branching tubes and examined the linear and nonlinear viscous cytoplasm flows in the tubes. We found that the nonlinearity in the cytoplasm׳s viscosity induces a novel type of symmetry breaking in the protoplasmic flow. We also show that symmetry breaking can play an important role in adaptive behaviors, namely, connection of behavioral modes implemented on different time scales and transportation of molecular signals from the front to the rear of the cell during cellular locomotion.

An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum

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.

Morphogenesis can be driven by properly parametrised mechanical feedback

A fundamental problem of morphogenesis is whether it presents itself as a succession of links that are each driven by its own specific cause-effect relationship, or whether all of the links can be embraced by a common law that is possible to formulate in physical terms. We suggest that a common biophysical background for most, if not all, morphogenetic processes is based upon feedback between mechanical stresses (MS) that are imposed to a given part of a developing embryo by its other parts and MS that are actively generated within that part. The latter are directed toward hyper-restoration (restoration with an overshoot) of the initial MS values. We show that under mechanical constraints imposed by other parts, these tendencies drive forth development. To provide specificity for morphogenetic reactions, this feedback should be modulated by long-term parameters and/or initial conditions that are set up by genetic factors. The experimental and model data related to this concept are reviewed.

Locomotive mechanism of Physarum plasmodia based on spatiotemporal analysis of protoplasmic streaming

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.

Intelligent behaviors of amoeboid movement based on complex dynamics of soft matter

We review how soft matter is self-organized to perform information processing at the cell level by examining the model organism Physarum plasmodium. The amoeboid organism, Physarum polycephalum, in the class of true slime molds, exhibits the intelligent behavior of foraging in complex situations. When placed in a maze with food sources at two exits, the organism develops tubular structures with its body which connect the food sources along the shortest path so that the rates of nutrient absorption and intracellular communication are maximized. This intelligent behavior results from the organism's control of a dynamic network through which mechanical and chemical information is transmitted. We review experimental studies that explore the development and adaptation of structures that make up the network. Recently a model of the dynamic network has been developed, and we review the formulation of this model and present some key results. The model captures the dynamics of existing networks, but it does not answer the question of how such networks form initially. To address the development of cell shape, we review existing mechanochemical models of the protoplasm of Physarum, present more general models of motile cells, and discuss how to adapt existing models to explore the development of intelligent networks in Physarum.

Physarum machines: encapsulating reaction-diffusion to compute spanning tree

The Physarum machine is a biological computing device, which employs plasmodium of Physarum polycephalum as an unconventional computing substrate. A reaction-diffusion computer is a chemical computing device that computes by propagating diffusive or excitation wave fronts. Reaction-diffusion computers, despite being computationally universal machines, are unable to construct certain classes of proximity graphs without the assistance of an external computing device. I demonstrate that the problem can be solved if the reaction-diffusion system is enclosed in a membrane with few 'growth points', sites guiding the pattern propagation. Experimental approximation of spanning trees by P. polycephalum slime mold demonstrates the feasibility of the approach. Findings provided advance theory of reaction-diffusion computation by enriching it with ideas of slime mold computation.

A mathematical model for adaptive transport network in path finding by true slime mold

We describe here a mathematical model of the adaptive dynamics of a transport network of the true slime mold Physarum polycephalum, an amoeboid organism that exhibits path-finding behavior in a maze. This organism possesses a network of tubular elements, by means of which nutrients and signals circulate through the plasmodium. When the organism is put in a maze, the network changes its shape to connect two exits by the shortest path. This process of path-finding is attributed to an underlying physiological mechanism: a tube thickens as the flux through it increases. The experimental evidence for this is, however, only qualitative. We constructed a mathematical model of the general form of the tube dynamics. Our model contains a key parameter corresponding to the extent of the feedback regulation between the thickness of a tube and the flux through it. We demonstrate the dependence of the behavior of the model on this parameter.

Mathematical model for rhythmic protoplasmic movement in the true slime mold

The plasmodium of the true slime mold Physarum polycephalum is a large amoeboid organism that displays "smart" behavior such as chemotaxis and the ability to solve mazes and geometrical puzzles. These amoeboid behaviors are based on the dynamics of the viscoelastic protoplasm and its biochemical rhythms. By incorporating both these aspects, we constructed a mathematical model for the dynamics of the organism as a first step towards understanding the relation between protoplasmic movement and its unusual abilities. We tested the validity of the model by comparing it with physiological observation. Our model reproduces fundamental characteristics of the spatio-temporal pattern of the rhythmic movement: (1) the antiphase oscillation between frontal tip and rear when the front is freely extending; (2) the asynchronous oscillation pattern when the front is not freely extending; and (3) the formation of protoplasmic mounds over a longer time scale. Both our model and physiological observation suggest that cell stiffness plays a primary role in plasmodial behaviors, in contrast to the conventional theory of coupled oscillator systems.

Interaction between cell shape and contraction pattern in the Physarum plasmodium

The relationship between cell shape and rhythmic contractile activity in the large amoeboid organism Physarum polycephalum was studied. The organism develops intricate networks of veins in which protoplasmic sol moved to and fro very regularly. When migrating on plain agar, the plasmodium extends like a sheet and develops dendritic veins toward the rear. After a particular stimulation, the vein organization changes into veinless or vein-network structures. In both structures, the mixing rate of the protoplasm, which is related to communication among contraction oscillators, decreased compared with that of the dendritic one. Accompanying these changes in vein structure, the spatio-temporal pattern of the rhythmic contraction changed into a small-structured pattern from a synchronized one. In the above process, cell shape affects the contraction pattern, but, conversely, the contraction pattern effects the cell shape. To demonstrate this, a phase difference in the rhythmic contraction was induced artificially by entraining the intrinsic rhythm to external temperature oscillations. New veins then formed along the direction parallel to the phase difference of the rhythm. Consequently, the vein organization of the cell interacts with the contractile activity to form a feedback loop in a mechanism of contraction pattern formation.

Inversely correlated cycles in speed and turning in an ameba: an oscillatory model of cell locomotion

Previous biophysical models of ameboid crawling have described cell movement in terms of a persistent random walk. Speed and orientation were treated in the latter model as independent and temporally homogeneous stochastic processes. We show here that, at least in the case of Dictyostelium discoideum, both speed control and reorientation processes involve a deterministic, periodic component. We also show that the processes are synchronized and negatively correlated, as was suggested by earlier findings. That is, increased turning correlates with periods of slow movement. Therefore, previous models are inconsistent with the behavior of cells. Using a heuristic approach, we have developed a mathematical model that describes the statistical properties of the cell's velocity and movement of its centroid. Our observations and the model are consistent with the phenomenological description of ameboid motility as a cyclic process of pseudopod extension and retraction.

Actin aggregation and embryonic epidermal wound healing

Recent experiments on the response of embryonic epidermis to wounding have revealed a cable of filamentous actin at the wound edge, which may be responsible for healing (Martin and Lewis 1991, 1992). We investigate the important question of how the cable forms as a response to wounding. We modify the mechanical model of Murray and Oster (1984) to investigate the post-wounding equilibrium in the epidermal sheet. We analyse the model in both one-dimensional and radially symmetric two-dimensional geometries, to determine the parameter domain in which a solution exists. We show that in both geometries the model solutions reflect the phenomenon of the actin cable for parameter values close to one edge of this domain. We interpret these results in terms of the relative rates of intracellular reorganization of actin and myosin, and thus suggest a possible mechanism for the formation of the actin cable.

Stress-induced alignment of actin filaments and the mechanics of cytogel

Experimental evidence suggests that anisotropic stress induces alignment of intracellular actin filaments. We develop a model for this phenomenon, which includes a parameter reflecting the sensitivity of the microfilament network to changes in the stress field. When applied to a uniform cell sheet at rest, the model predicts that for sufficiently large values of the sensitivity parameter, all the actin filaments will spontaneously align in a single direction. Stress alignment can also be caused by a change in external conditions, and as an example of this we apply our model to the initial response of embryonic epidermis to wounding. Our solutions in this case are able to reflect the actin cable that has been found at the wound edge in recent experiments; the cable consists of microfilaments aligned with stress at the wound boundary of the epithelium. These applications suggest that stress-induced alignment of actin filaments could play a key role in some biological systems. This is the first attempt to include the alignment phenomenon in a mechanical model of cytogel.

Calcium wave propagation by calcium-induced calcium release: an unusual excitable system

We discuss in detail the behaviour of a model, proposed by Goldbeter et al. (1990. Proc. natn. Acad. Sci. 87, 1461-1465), for intracellular calcium wave propagation by calcium-induced calcium release, focusing our attention on excitability and the propagation of waves in one spatial dimension. The model with no diffusion behaves like a generic excitable system, and threshold behaviour, excitability and oscillations can be understood within this general framework. However, when diffusion is included, the model no longer behaves like a generic excitable system; the fast and slow variables are not distinct and previous results on excitable systems do not necessarily apply. We consider a piecewise linear simplification of the model, and construct travelling pulse and periodic plane wave solutions to the simplified model. The analogous behaviour in the full model is studied numerically. Goldbeter's model for calcium-induced calcium release is an excitable system of a type not previously studied in detail.

Analysis of dynamic and stationary pattern formation in the cell cortex

We study a sol-gel mechanochemical model for cellular cytoplasm. Using conservation equations and a force balance equation, we derive equations for the sol-gel dynamics. Regular perturbation analysis suggests the growth of patterns which may be either dynamic or stationary, depending on parameter values. Nonlinear analysis, which indicates that these patterns remain bounded, is confirmed by numerically solving the mechanochemical equations. We use these analytical and numerical results to model two different biological problems: the dynamic formation of filopodia in nerve growth cones, and the growth of microvilli in epithelial cells.

Model of the Ca2+ oscillator for shuttle streaming in Physarum polycephalum

We propose a mechanism for the cytoplasmic Ca++ oscillator which is thought to power shuttle streaming in strands of the slime-mold Physarum polycephalum. The mechanism uses a phosphorylation-dephosphorylation cycle of myosin light chain kinase. This kinase is bistable if the kinase phosphorylation chain, through adenylate cyclase and cAMP, is activated by calcium. Relaxation oscillations can then occur if calcium is exchanged between the cytoplasm and internal vacuoles known to exist in physarum. As contractile activity in physarum myosin is inhibited by calcium, this model can give calcium oscillations 180 degrees out of phase with actin filament tension as observed. Oscillations of ATP concentration are correctly predicted to be in phase with the tension, provided the actomyosin cycling rate is comparable with ATPase rates for phosphorylation of the myosin light chain and its kinase.

A continuum model of contraction waves and protoplasm streaming in strands of Physarum plasmodium

We present a mathematical model for continuously distributed mechanochemical autooscillations (autowaves) in a protoplasmic strand of Physarum polycephalum. The model is based on a hypothesis of local positive feedback between deformation and contraction of the contractile apparatus. This feedback is mediated through a cell regulatory system whose kinetics involves coupling to mechanical strain. Mathematical analysis and computer simulations have demonstrated that the solutions of the model agree quantitatively with the available experimental data. In particular, hydrodynamic interaction alone, between different sections of the strand via the streaming endoplasm, is capable of inducing the characteristic contractile behavior.

On the elasticity of cytoskeletal networks

Models relating to the gelation and elasticity of complex cytoskeletal networks are formulated and investigated. Kinetic equations for reversible elongation of nucleated actin filaments are analyzed when the filaments are acted upon by capping proteins and cross-linking factors. Analytical expressions are obtained that relate the low frequency elastic shear modulus of a network, G, to chain growth kinetics, the number of nucleation sites, monomer concentration, and the amount of capping and cross-linking protein. Elasticity curves that relate G to such factors as the association constant for cross-linking are derived and then used to determine solation-gelation phase contours.

Dependence of the mechanical properties of actin/alpha-actinin gels on deformation rate

The cortical cytoplasm, including the cleavage furrow, is largely composed of a network of actin filaments that is rigid even as it is extensively deformed during cytokinesis. Here we address the question of how actin-filament networks such as those in the cortex can be simultaneously rigid (solid-like) and fluid-like. Conventional explanations are that actin filaments rearrange by some combination of depolymerization and repolymerization; fragmentation and annealing of filaments; and inactivation and reestablishment of crosslinks between filaments. We describe the mechanical properties of a model system consisting of actin filaments and Acanthamoeba alpha-actinin, one of several actin crosslinking proteins found in amoeba and other cells. The results suggest another molecular mechanism that may account for the paradoxical mechanical properties of the cortex. When deformed rapidly, these mixtures are 40 times more rigid than actin filaments without alpha-actinin, but when deformed slowly these mixtures were indistinguishable from filaments alone. These time-dependent mechanical properties can be explained by multiple, rapidly rearranging alpha-actinin crosslinks between the actin filaments, a mechanism proposed by Frey-Wyssling to account for the behaviour of cytoplasm long before the discovery of cytoplasmic actin or alpha-actinin. If other actin-filament crosslinking proteins behave like Acanthamoeba alpha-actinin, this mechanism may explain how the cortex recoils elastically from small rapid insults but deforms extensively when minute forces are applied over long periods of time.

Cortical activity in vertebrate eggs. I: The activation waves

We present a physical model for the propagation of chemical and mechanical waves on the surface of vertebrate eggs. As a first step we analyzed the propagation of the calcium wave observed to sweep over the surface of the Medaka egg (Gilkey et al., 1978). It has been assumed that this wave is driven by a mechanism of calcium-stimulated-calcium-release. By formulating this hypothesis mathematically we can use the observed wavefront data to obtain a map of cortical reactivity. This map indicates a gradient of reactivity along the egg: highest in the animal hemisphere and tapering off towards the vegetal hemisphere. The cortex of Xenopus eggs is also capable of propagating a calcium wave (Busa & Nuccitelli, 1985). At about the same time a wave of expansion followed by a wave of contraction sweeps across the egg surface (Takeichi et al., 1984). We have proposed a mechanism for this wave pair based on the physical chemistry of actomyosin gels. The calcium wave activates solation factors which sever some of the actin chains which leads to an osmotic swelling of the gel. Calcium also activates the contractile machinery of the actomyosin system which causes the gel to contract. The contraction lags the swelling because of the nature of the kinetics: solation and swelling is a more rapid process than contraction. By writing the equations for gel expansion and contraction we can mimic the mechanical and chemical wave propagation by a computer simulation. If the model is correct this provides a method for using the waves as a diagnostic of the mechanochemical properties of the egg cortex.

Consequences of chemosensory phenomena for leukocyte chemotactic orientation

The stochastic nature of cell surface receptor-ligand binding is known to limit the accuracy of detection of chemoattractant gradients by leukocytes, thus limiting the orientation ability that is crucial to the chemotactic response in host defense. The probabilistic cell orientation model of Lauffenburger is extended here to assess the consequences of recently discovered receptor phenomena: "down-regulation" of total surface receptor number, spatial asymmetry of surface receptors, and existence of a higher-affinity receptor subpopulation. In general, a reduction in orientation accuracy is predicted by inclusion of these phenomena. An orientation signal based on a simple model of chemosensory adaptation (i.e., a spatial difference in relative receptor occupancy) is found to be functionally different from the signal suggested by an experimental correlation (i.e., a spatial difference in absolute receptor occupancy). However, in the context of receptor "signal noise," the signal based on adaptation yields predictions in better qualitative agreement with the experimental orientation data of Zigmond. From this cell orientation model we can estimate the effective time-averaging period required for noise diminution to a level allowing orientation predictions to match observed levels. This time-averaging period presumably reflects the time constant for receptor signal transduction and locomotory response.