Theoretical physics of biological development

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The steroid hormone 20-hydroxy-ecdysone (20E) promotes proliferation in Drosophila wing precursors at low titer but triggers proliferation arrest at high doses. Remarkably, wing precursors proliferate normally in the complete absence of the 20E receptor, suggesting that low-level 20E promotes proliferation by overriding the default anti-proliferative activity of the receptor. By contrast, 20E needs its receptor to arrest proliferation. Dose-response RNA sequencing (RNA-seq) analysis of ex vivo cultured wing precursors identifies genes that are quantitatively activated by 20E across the physiological range, likely comprising positive modulators of proliferation and other genes that are only activated at high doses. We suggest that some of these "high-threshold" genes dominantly suppress the activity of the pro-proliferation genes. We then show mathematically and with synthetic reporters that combinations of basic regulatory elements can recapitulate the behavior of both types of target genes. Thus, a relatively simple genetic circuit can account for the bimodal activity of this hormone.

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Segmenting cells within cellular aggregates in 3D is a growing challenge in cell biology, due to improvements in capacity and accuracy of microscopy techniques. Here we describe a pipeline to segment images of cell aggregates in 3D. The pipeline combines neural network segmentations with active meshes. We apply our segmentation method to cultured mouse mammary duct organoids imaged over 24 hours with oblique plane microscopy, a high-throughput light-sheet fluorescence microscopy technique. We show that our method can also be applied to images of mouse embryonic stem cells imaged with a spinning disc microscope. We segment individual cells based on nuclei and cell membrane fluorescent markers, and track cells over time. We describe metrics to quantify the quality of the automated segmentation. Our segmentation pipeline involves a Fiji plugin which implement active meshes deformation and allows a user to create training data, automatically obtain segmentation meshes from original image data or neural network prediction, and manually curate segmentation data to identify and correct mistakes. Our active meshes-based approach facilitates segmentation postprocessing, correction, and integration with neural network prediction.

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Shape transformations of epithelial tissues in three dimensions, which are crucial for embryonic development or in vitro organoid growth, can result from active forces generated within the cytoskeleton of the epithelial cells. How the interplay of local differential tensions with tissue geometry and with external forces results in tissue-scale morphogenesis remains an open question. Here, we describe epithelial sheets as active viscoelastic surfaces and study their deformation under patterned internal tensions and bending moments. In addition to isotropic effects, we take into account nematic alignment in the plane of the tissue, which gives rise to shape-dependent, anisotropic active tensions and bending moments. We present phase diagrams of the mechanical equilibrium shapes of pre-patterned closed shells and explore their dynamical deformations. Our results show that a combination of nematic alignment and gradients in internal tensions and bending moments is sufficient to reproduce basic building blocks of epithelial morphogenesis, including fold formation, budding, neck formation, flattening, and tubulation.

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We introduce a modelling and simulation framework for cell aggregates in three dimensions based on interacting active surfaces. Cell mechanics is captured by a physical description of the acto-myosin cortex that includes cortical flows, viscous forces, active tensions, and bending moments. Cells interact with each other via short-range forces capturing the effect of adhesion molecules. We discretise the model equations using a finite element method, and provide a parallel implementation in C++. We discuss examples of application of this framework to small and medium-sized aggregates: we consider the shape and dynamics of a cell doublet, a planar cell sheet, and a growing cell aggregate. This framework opens the door to the systematic exploration of the cell to tissue-scale mechanics of cell aggregates, which plays a key role in the morphogenesis of embryos and organoids.

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Collective motions of epithelial cells are essential for morphogenesis. Tissues elongate, contract, flow, and oscillate, thus sculpting embryos. These tissue level dynamics are known, but the physical mechanisms at the cellular level are unclear. Here, we demonstrate that a single epithelial monolayer of MDCK cells can exhibit two types of local tissue kinematics, pulsations and long range coherent flows, characterized by using quantitative live imaging. We report that these motions can be controlled with internal and external cues such as specific inhibitors and substrate friction modulation. We demonstrate the associated mechanisms with a unified vertex model. When cell velocity alignment and random diffusion of cell polarization are comparable, a pulsatile flow emerges whereas tissue undergoes long-range flows when velocity alignment dominates which is consistent with cytoskeletal dynamics measurements. We propose that environmental friction, acto-myosin distributions, and cell polarization kinetics are important in regulating dynamics of tissue morphogenesis.

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We derive a fully covariant theory of the hydrodynamics of nematic and polar active surfaces, subjected to internal and external forces and torques. We study the symmetries of polar and nematic surfaces and find that in addition to five different types of in-plane isotropic surfaces, polar and nematic surfaces can be classified into five polar, two pseudopolar, five nematic and two pseudonematic types of surfaces. We give examples of physical realisations of the different types of surfaces we have identified. We obtain expressions for the equilibrium tensions, moments, and external forces and torques acting on a passive polar or nematic surface. We calculate the entropy production rate using the framework of thermodynamics close to equilibrium and find constitutive equations for polar and nematic active surfaces with different symmetries. We study the instabilities of a confined flat planar-chiral polar active layer and of a confined deformable polar active surface with broken up-down symmetry.

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Proper orientation of the mitotic spindle plays a crucial role in embryos, during tissue development, and in adults, where it functions to dissipate mechanical stress to maintain tissue integrity and homeostasis. While mitotic spindles have been shown to reorient in response to external mechanical stresses, the subcellular cues that mediate spindle reorientation remain unclear. Here, we used a combination of optogenetics and computational modeling to investigate how mitotic spindles respond to inhomogeneous tension within the actomyosin cortex. Strikingly, we found that the optogenetic activation of RhoA only influences spindle orientation when it is induced at both poles of the cell. Under these conditions, the sudden local increase in cortical tension induced by RhoA activation reduces pulling forces exerted by cortical regulators on astral microtubules. This leads to a perturbation of the balance of torques exerted on the spindle, which causes it to rotate. Thus, spindle rotation in response to mechanical stress is an emergent phenomenon arising from the interaction between the spindle positioning machinery and the cell cortex.

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Entropy production plays a fundamental role in the study of nonequilibrium systems by offering a quantitative handle on the degree of time-reversal symmetry breaking. It depends crucially on the degree of freedom considered as well as on the scale of description. How the entropy production at one resolution of the degrees of freedom is related to the entropy production at another resolution is a fundamental question which has recently attracted interest. This relationship is of particular relevance to coarse-grained and continuum descriptions of a given phenomenon. In this work, we derive the scaling of the entropy production under iterative coarse graining on the basis of the correlations of the underlying microscopic transition rates for noninteracting particles in active disordered media. Our approach unveils a natural criterion to distinguish equilibrium-like and genuinely nonequilibrium macroscopic phenomena based on the sign of the scaling exponent of the entropy production per mesostate.

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As organs and tissues approach their normal size during development or regeneration, growth slows down, and cell proliferation progressively comes to a halt. Among the various processes suggested to contribute to growth termination, mechanical feedback, perhaps via adherens junctions, has been suggested to play a role. However, since adherens junctions are only present in a narrow plane of the subapical region, other structures are likely needed to sense mechanical stresses along the apical-basal (A-B) axis, especially in a thick pseudostratified epithelium. This could be achieved by nuclei, which have been implicated in mechanotransduction in tissue culture. In addition, mechanical constraints imposed by nuclear crowding and spatial confinement could affect interkinetic nuclear migration (IKNM), which allows G2 nuclei to reach the apical surface, where they normally undergo mitosis. To explore how mechanical constraints affect IKNM, we devised an individual-based model that treats nuclei as deformable objects constrained by the cell cortex and the presence of other nuclei. The model predicts changes in the proportion of cell-cycle phases during growth, which we validate with the cell-cycle phase reporter FUCCI (Fluorescent Ubiquitination-based Cell Cycle Indicator). However, this model does not preclude indefinite growth, leading us to postulate that nuclei must migrate basally to access a putative basal signal required for S phase entry. With this refinement, our updated model accounts for the observed progressive slowing down of growth and explains how pseudostratified epithelia reach a stereotypical thickness upon completion of growth.

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During development, multicellular organisms undergo stereotypical patterns of tissue growth in space and time. How developmental growth is orchestrated remains unclear, largely due to the difficulty of observing and quantitating this process in a living organism. Drosophila histoblast nests are small clusters of progenitor epithelial cells that undergo extensive growth to give rise to the adult abdominal epidermis and are amenable to live imaging. Our quantitative analysis of histoblast proliferation and tissue mechanics reveals that tissue growth is driven by cell divisions initiated through basal extracellular matrix degradation by matrix metalloproteases secreted by the neighboring larval epidermal cells. Laser ablations and computational simulations show that tissue mechanical tension does not decrease as the histoblasts fill the abdominal epidermal surface. During tissue growth, the histoblasts display oscillatory cell division rates until growth termination occurs through the rapid emergence of G0/G1 arrested cells, rather than a gradual increase in cell-cycle time as observed in other systems such as the Drosophila wing and mouse postnatal epidermis. Different developing tissues can therefore achieve their final size using distinct growth termination strategies. Thus, adult abdominal epidermal development is characterized by changes in the tissue microenvironment and a rapid exit from the cell cycle.

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The cell cortex is a highly dynamic network of cytoskeletal filaments in which motor proteins induce active cortical stresses which in turn drive dynamic cellular processes such as cell motility, furrow formation or cytokinesis during cell division. Here, we develop a three-dimensional computational model of a cell cortex in the viscous limit including active cortical flows. Combining active gel and thin shell theory, we base our computational tool directly on the force balance equations for the velocity field on a discretized and arbitrarily deforming cortex. Since our method is based on the general force balance equations, it can easily be extended to more complex biological dependencies in terms of the constitutive laws or a dynamic coupling to a suspending fluid. We validate our algorithm by investigating the formation of a cleavage furrow on a biological cell immersed in a passive outer fluid, where we successfully compare our results to axi-symmetric simulations. We then apply our fully three-dimensional algorithm to fold formation and to study furrow formation under the influence of non-axisymmetric disturbances such as external shear. We report a reorientation mechanism by which the cell autonomously realigns its axis perpendicular to the furrow plane thus contributing to the robustness of cell division under realistic environmental conditions.

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Lumen formation plays an essential role in the morphogenesis of tissues during development. Here we review the physical principles that play a role in the growth and coarsening of lumens. Solute pumping by the cell, hydraulic flows driven by differences of osmotic and hydrostatic pressures, balance of forces between extracellular fluids and cell-generated cytoskeletal forces, and electro-osmotic effects have been implicated in determining the dynamics and steady-state of lumens. We use the framework of linear irreversible thermodynamics to discuss the relevant force, time and length scales involved in these processes. We focus on order of magnitude estimates of physical parameters controlling lumen formation and coarsening.

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We introduce a perturbative method to calculate all moments of the first passage time distribution in stochastic one-dimensional processes which are subject to both white and colored noise. This class of non-Markovian processes is at the center of the study of thermal active matter, that is self-propelled particles subject to diffusion. The perturbation theory about the Markov process considers the effect of self-propulsion to be small compared to that of thermal fluctuations. To illustrate our method, we apply it to the case of active thermal particles (i) in a harmonic trap and (ii) on a ring. For both we calculate the first-order correction of the moment-generating function of first passage times, and thus to all its moments. Our analytical results are compared to numerics.

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We present a new folded dual-view oblique plane microscopy (OPM) technique termed dOPM that enables two orthogonal views of the sample to be obtained by translating a pair of tilted mirrors in refocussing space. Using a water immersion 40× 1.15 NA primary objective, deconvolved image volumes of 200 nm beads were measured to have full width at half maxima (FWHM) of 0.35 ± 0.04 µm and 0.39 ± 0.02 µm laterally and 0.81 ± 0.07 µm axially. The measured z-sectioning value was 1.33 ± 0.45 µm using light-sheet FWHM in the frames of the two views of 4.99 ± 0.58 µm and 4.89 ± 0.63 µm. To qualitatively demonstrate that the system can reduce shadow artefacts while providing a more isotropic resolution, a multi-cellular spheroid approximately 100 µm in diameter was imaged.

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Epithelial tissues of the developing embryos elongate by different mechanisms, such as neighbor exchange, cell elongation, and oriented cell division. Since autonomous tissue self-organization is influenced by external cues such as morphogen gradients or neighboring tissues, it is difficult to distinguish intrinsic from directed tissue behavior. The mesoscopic processes leading to the different mechanisms remain elusive. Here, we study the spontaneous elongation behavior of spreading circular epithelial colonies in vitro. By quantifying deformation kinematics at multiple scales, we report that global elongation happens primarily due to cell elongations, and its direction correlates with the anisotropy of the average cell elongation. By imposing an external time-periodic stretch, the axis of this global symmetry breaking can be modified and elongation occurs primarily due to orientated neighbor exchange. These different behaviors are confirmed using a vertex model for collective cell behavior, providing a framework for understanding autonomous tissue elongation and its origins.

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Morphogen gradients provide positional information during development. To uncover the minimal requirements for morphogen gradient formation, we have engineered a synthetic morphogen in Drosophila wing primordia. We show that an inert protein, green fluorescent protein (GFP), can form a detectable diffusion-based gradient in the presence of surface-associated anti-GFP nanobodies, which modulate the gradient by trapping the ligand and limiting leakage from the tissue. We next fused anti-GFP nanobodies to the receptors of Dpp, a natural morphogen, to render them responsive to extracellular GFP. In the presence of these engineered receptors, GFP could replace Dpp to organize patterning and growth in vivo. Concomitant expression of glycosylphosphatidylinositol (GPI)-anchored nonsignaling receptors further improved patterning, to near-wild-type quality. Theoretical arguments suggest that GPI anchorage could be important for these receptors to expand the gradient length scale while at the same time reducing leakage.

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Proliferating animal cells are able to orient their mitotic spindles along their interphase cell axis, setting up the axis of cell division, despite rounding up as they enter mitosis. This has previously been attributed to molecular memory and, more specifically, to the maintenance of adhesions and retraction fibers in mitosis [1, 2, 3, 4, 5, 6], which are thought to act as local cues that pattern cortical Gαi, LGN, and nuclear mitotic apparatus protein (NuMA) [3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. This cortical machinery then recruits and activates Dynein motors, which pull on astral microtubules to position the mitotic spindle. Here, we reveal a dynamic two-way crosstalk between the spindle and cortical motor complexes that depends on a Ran-guanosine triphosphate (GTP) signal [12], which is sufficient to drive continuous monopolar spindle motion independently of adhesive cues in flattened human cells in culture. Building on previous work [1, 12, 19, 20, 21, 22, 23], we implemented a physical model of the system that recapitulates the observed spindle-cortex interactions. Strikingly, when this model was used to study spindle dynamics in cells entering mitosis, the chromatin-based signal was found to preferentially clear force generators from the short cell axis, so that cortical motors pulling on astral microtubules align bipolar spindles with the interphase long cell axis, without requiring a fixed cue or a physical memory of interphase shape. Thus, our analysis shows that the ability of chromatin to pattern the cortex during the process of mitotic rounding is sufficient to translate interphase shape into a cortical pattern that can be read by the spindle, which then guides the axis of cell division.

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During development, cell-generated forces induce tissue-scale deformations to shape the organism [1,2]. The pattern and extent of these deformations depend not solely on the temporal and spatial profile of the generated force fields but also on the mechanical properties of the tissues that the forces act on. It is thus conceivable that, much like the cell-generated forces, the mechanical properties of tissues are modulated during development in order to drive morphogenesis toward specific developmental endpoints. Although many approaches have recently emerged to assess effective mechanical parameters of tissues [3-8], they could not quantitatively relate spatially localized force induction to tissue-scale deformations in vivo. Here, we present a method that overcomes this limitation. Our approach is based on the application of controlled forces on a single microparticle embedded in an individual cell of an embryo. Combining measurements of bead displacement with the analysis of induced deformation fields in a continuum mechanics framework, we quantify material properties of the tissue and follow their changes over time. In particular, we uncover a rapid change in tissue response occurring during Drosophila cellularization, resulting from a softening of the blastoderm and an increase of external friction. We find that the microtubule cytoskeleton is a major contributor to epithelial mechanics at this stage. We identify developmentally controlled modulations in perivitelline spacing that can account for the changes in friction. Overall, our method allows for the measurement of key mechanical parameters governing tissue-scale deformations and flows occurring during morphogenesis.

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Tubular epithelia are a basic building block of organs and a common site of cancer occurrence1-4. During tumorigenesis, transformed cells overproliferate and epithelial architecture is disrupted. However, the biophysical parameters that underlie the adoption of abnormal tumour tissue shapes are unknown. Here we show in the pancreas of mice that the morphology of epithelial tumours is determined by the interplay of cytoskeletal changes in transformed cells and the existing tubular geometry. To analyse the morphological changes in tissue architecture during the initiation of cancer, we developed a three-dimensional whole-organ imaging technique that enables tissue analysis at single-cell resolution. Oncogenic transformation of pancreatic ducts led to two types of neoplastic growth: exophytic lesions that expanded outwards from the duct and endophytic lesions that grew inwards to the ductal lumen. Myosin activity was higher apically than basally in wild-type cells, but upon transformation this gradient was lost in both lesion types. Three-dimensional vertex model simulations and a continuum theory of epithelial mechanics, which incorporate the cytoskeletal changes observed in transformed cells, indicated that the diameter of the source epithelium instructs the morphology of growing tumours. Three-dimensional imaging revealed that-consistent with theory predictions-small pancreatic ducts produced exophytic growth, whereas large ducts deformed endophytically. Similar patterns of lesion growth were observed in tubular epithelia of the liver and lung; this finding identifies tension imbalance and tissue curvature as fundamental determinants of epithelial tumorigenesis.

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Tissue morphogenesis is driven by mechanical forces that elicit changes in cell size, shape and motion. The extent by which forces deform tissues critically depends on the rheological properties of the recipient tissue. Yet, whether and how dynamic changes in tissue rheology affect tissue morphogenesis and how they are regulated within the developing organism remain unclear. Here, we show that blastoderm spreading at the onset of zebrafish morphogenesis relies on a rapid, pronounced and spatially patterned tissue fluidization. Blastoderm fluidization is temporally controlled by mitotic cell rounding-dependent cell-cell contact disassembly during the last rounds of cell cleavages. Moreover, fluidization is spatially restricted to the central blastoderm by local activation of non-canonical Wnt signalling within the blastoderm margin, increasing cell cohesion and thereby counteracting the effect of mitotic rounding on contact disassembly. Overall, our results identify a fluidity transition mediated by loss of cell cohesion as a critical regulator of embryo morphogenesis.

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Cell division and death can be regulated by the mechanical forces within a tissue. We study the consequences for the stability and roughness of a propagating interface by analyzing a model of mechanically regulated tissue growth in the regime of small driving forces. For an interface driven by homeostatic pressure imbalance or leader-cell motility, long and intermediate-wavelength instabilities arise, depending, respectively, on an effective viscosity of cell number change, and on substrate friction. A further mechanism depends on the strength of directed motility forces acting in the bulk. We analyze the fluctuations of a stable interface subjected to cell-level stochasticity, and find that mechanical feedback can help preserve reproducibility at the tissue scale. Our results elucidate mechanisms that could be important for orderly interface motion in developing tissues.

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During epithelial contraction, cells generate forces to constrict their surface and, concurrently, fine-tune the length of their adherens junctions to ensure force transmission. While many studies have focused on understanding force generation, little is known on how junctional length is controlled. Here, we show that, during amnioserosa contraction in Drosophila dorsal closure, adherens junctions reduce their length in coordination with the shrinkage of apical cell area, maintaining a nearly constant junctional straightness. We reveal that junctional straightness and integrity depend on the endocytic machinery and on the mechanosensitive activity of the actomyosin cytoskeleton. On one hand, upon junctional stretch and decrease in E-cadherin density, actomyosin relocalizes from the medial area to the junctions, thus maintaining junctional integrity. On the other hand, when junctions have excess material and ruffles, junction removal is enhanced, and high junctional straightness and tension are restored. These two mechanisms control junctional length and integrity during morphogenesis.

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Epithelial folding transforms simple sheets of cells into complex three-dimensional tissues and organs during animal development. Epithelial folding has mainly been attributed to mechanical forces generated by an apically localized actomyosin network, however, contributions of forces generated at basal and lateral cell surfaces remain largely unknown. Here we show that a local decrease of basal tension and an increased lateral tension, but not apical constriction, drive the formation of two neighboring folds in developing Drosophila wing imaginal discs. Spatially defined reduction of extracellular matrix density results in local decrease of basal tension in the first fold; fluctuations in F-actin lead to increased lateral tension in the second fold. Simulations using a 3D vertex model show that the two distinct mechanisms can drive epithelial folding. Our combination of lateral and basal tension measurements with a mechanical tissue model reveals how simple modulations of surface and edge tension drive complex three-dimensional morphological changes.

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Tissue shape is often established early in development and needs to be scaled isotropically during growth. However, the cellular contributors and ways by which cells interact tissue-wide to enable coordinated isotropic tissue scaling are not yet understood. Here, we follow cell and tissue shape changes in the zebrafish retinal neuroepithelium, which forms a cup with a smooth surface early in development and maintains this architecture as it grows. By combining 3D analysis and theory, we show how a global increase in cell height can maintain tissue shape during growth. Timely cell height increase occurs concurrently with a non-cell-autonomous actin redistribution. Blocking actin redistribution and cell height increase perturbs isotropic scaling and leads to disturbed, folded tissue shape. Taken together, our data show how global changes in cell shape enable isotropic growth of the developing retinal neuroepithelium, a concept that could also apply to other systems.

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Reconstitution of a Hedgehog morphogen gradient in vitro and in silico reveals the architectural features of the signal transduction pathway that ensure rapid formation of a robust signalling gradient.

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Epithelial tissues can elongate in two dimensions by polarized cell intercalation, oriented cell division, or cell shape change, owing to local or global actomyosin contractile forces acting in the plane of the tissue. In addition, epithelia can undergo morphogenetic change in three dimensions. We show that elongation of the wings and legs of Drosophila involves a columnar-to-cuboidal cell shape change that reduces cell height and expands cell width. Remodeling of the apical extracellular matrix by the Stubble protease and basal matrix by MMP1/2 proteases induces wing and leg elongation. Matrix remodeling does not occur in the haltere, a limb that fails to elongate. Limb elongation is made anisotropic by planar polarized Myosin-II, which drives convergent extension along the proximal-distal axis. Subsequently, Myosin-II relocalizes to lateral membranes to accelerate columnar-to-cuboidal transition and isotropic tissue expansion. Thus, matrix remodeling induces dynamic changes in actomyosin contractility to drive epithelial morphogenesis in three dimensions.

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In embryonic development or tumor evolution, cells often migrate collectively within confining tracks defined by their microenvironment 1,2. In some of these situations, the displacements within a cell strand are antiparallel 3, giving rise to shear flows. However, the mechanisms underlying these spontaneous flows remain poorly understood. Here, we show that an ensemble of spindle-shaped cells plated in a well-defined stripe spontaneously develop a shear flow whose characteristics depend on the width of the stripe. On wide stripes, the cells self-organize in a nematic phase with a director at a well-defined angle with the stripe's direction, and develop a shear flow close to the stripe's edges. However, on stripes narrower than a critical width, the cells perfectly align with the stripe's direction and the net flow vanishes. A hydrodynamic active gel theory provides an understanding of these observations and identifies the transition between the non-flowing phase oriented along the stripe and the tilted phase exhibiting shear flow as a Fréedericksz transition driven by the activity of the cells. This physical theory is grounded in the active nature of the cells and based on symmetries and conservation laws, providing a generic mechanism to interpret in vivo antiparallel cell displacements.

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We review the general hydrodynamic theory of active soft materials that is motivated in particular by biological matter. We present basic concepts of irreversible thermodynamics of spatially extended multicomponent active systems. Starting from the rate of entropy production, we identify conjugate thermodynamic fluxes and forces and present generic constitutive equations of polar active fluids and active gels. We also discuss angular momentum conservation which plays a role in the the physics of active chiral gels. The irreversible thermodynamics of active gels provides a general framework to discuss the physics that underlies a wide variety of biological processes in cells and in multicellular tissues.

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Under conditions of homeostasis, dynamic changes in the length of individual adherens junctions (AJs) provide epithelia with the fluidity required to maintain tissue integrity in the face of intrinsic and extrinsic forces. While the contribution of AJ remodeling to developmental morphogenesis has been intensively studied, less is known about AJ dynamics in other circumstances. Here, we study AJ dynamics in an epithelium that undergoes a gradual increase in packing order, without concomitant large-scale changes in tissue size or shape. We find that neighbor exchange events are driven by stochastic fluctuations in junction length, regulated in part by junctional actomyosin. In this context, the developmental increase of isotropic junctional actomyosin reduces the rate of neighbor exchange, contributing to tissue order. We propose a model in which the local variance in tension between junctions determines whether actomyosin-based forces will inhibit or drive the topological transitions that either refine or deform a tissue.

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We derive a fully covariant theory of the mechanics of active surfaces. This theory provides a framework for the study of active biological or chemical processes at surfaces, such as the cell cortex, the mechanics of epithelial tissues, or reconstituted active systems on surfaces. We introduce forces and torques acting on a surface, and derive the associated force balance conditions. We show that surfaces with in-plane rotational symmetry can have broken up-down, chiral, or planar-chiral symmetry. We discuss the rate of entropy production in the surface and write linear constitutive relations that satisfy the Onsager relations. We show that the bending modulus, the spontaneous curvature, and the surface tension of a passive surface are renormalized by active terms. Finally, we identify active terms which are not found in a passive theory and discuss examples of shape instabilities that are related to active processes in the surface.

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Animal cell shape is largely determined by the cortex, a thin actin network underlying the plasma membrane in which myosin-driven stresses generate contractile tension. Tension gradients result in local contractions and drive cell deformations. Previous cortical tension regulation studies have focused on myosin motors. Here, we show that cortical actin network architecture is equally important. First, we observe that actin cortex thickness and tension are inversely correlated during cell-cycle progression. We then show that the actin filament length regulators CFL1, CAPZB and DIAPH1 regulate mitotic cortex thickness and find that both increasing and decreasing thickness decreases tension in mitosis. This suggests that the mitotic cortex is poised close to a tension maximum. Finally, using a computational model, we identify a physical mechanism by which maximum tension is achieved at intermediate actin filament lengths. Our results indicate that actin network architecture, alongside myosin activity, is key to cell surface tension regulation.

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Tissue morphogenesis requires the collective, coordinated motion and deformation of a large number of cells. Vertex model simulations for tissue mechanics have been developed to bridge the scales between force generation at the cellular level and tissue deformation and flows. We review here various formulations of vertex models that have been proposed for describing tissues in two and three dimensions. We discuss a generic formulation using a virtual work differential, and we review applications of vertex models to biological morphogenetic processes. We also highlight recent efforts to obtain continuum theories of tissue mechanics, which are effective, coarse-grained descriptions of vertex models.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

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During embryonic development, mechanical forces are essential for cellular rearrangements driving tissue morphogenesis. Here, we show that in the early zebrafish embryo, friction forces are generated at the interface between anterior axial mesoderm (prechordal plate, ppl) progenitors migrating towards the animal pole and neurectoderm progenitors moving in the opposite direction towards the vegetal pole of the embryo. These friction forces lead to global rearrangement of cells within the neurectoderm and determine the position of the neural anlage. Using a combination of experiments and simulations, we show that this process depends on hydrodynamic coupling between neurectoderm and ppl as a result of E-cadherin-mediated adhesion between those tissues. Our data thus establish the emergence of friction forces at the interface between moving tissues as a critical force-generating process shaping the embryo.

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In this article, we propose a general framework to study the dynamics and topology of cellular networks that capture the geometry of cell packings in two-dimensional tissues. Such epithelia undergo large-scale deformation during morphogenesis of a multicellular organism. Large-scale deformations emerge from many individual cellular events such as cell shape changes, cell rearrangements, cell divisions, and cell extrusions. Using a triangle-based representation of cellular network geometry, we obtain an exact decomposition of large-scale material deformation. Interestingly, our approach reveals contributions of correlations between cellular rotations and elongation as well as cellular growth and elongation to tissue deformation. Using this triangle method, we discuss tissue remodeling in the developing pupal wing of the fly Drosophila melanogaster.

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We present a hydrodynamic theory to describe shear flows in developing epithelial tissues. We introduce hydrodynamic fields corresponding to state properties of constituent cells as well as a contribution to overall tissue shear flow due to rearrangements in cell network topology. We then construct a generic linear constitutive equation for the shear rate due to topological rearrangements and we investigate a novel rheological behaviour resulting from memory effects in the tissue. We identify two distinct active cellular processes: generation of active stress in the tissue, and actively driven topological rearrangements. We find that these two active processes can produce distinct cellular and tissue shape changes, depending on boundary conditions applied on the tissue. Our findings have consequences for the understanding of tissue morphogenesis during development.

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Embryo morphogenesis relies on highly coordinated movements of different tissues. However, remarkably little is known about how tissues coordinate their movements to shape the embryo. In zebrafish embryogenesis, coordinated tissue movements first become apparent during "doming," when the blastoderm begins to spread over the yolk sac, a process involving coordinated epithelial surface cell layer expansion and mesenchymal deep cell intercalations. Here, we find that active surface cell expansion represents the key process coordinating tissue movements during doming. By using a combination of theory and experiments, we show that epithelial surface cells not only trigger blastoderm expansion by reducing tissue surface tension, but also drive blastoderm thinning by inducing tissue contraction through radial deep cell intercalations. Thus, coordinated tissue expansion and thinning during doming relies on surface cells simultaneously controlling tissue surface tension and radial tissue contraction.

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Cytokinesis in eukaryotic cells is often accompanied by actomyosin cortical flow. Over 30 years ago, Borisy and White proposed that cortical flow converging upon the cell equator compresses the actomyosin network to mechanically align actin filaments. However, actin filaments also align via search-and-capture, and to what extent compression by flow or active alignment drive furrow formation remains unclear. Here, we quantify the dynamical organization of actin filaments at the onset of ring assembly in the C. elegans zygote, and provide a framework for determining emergent actomyosin material parameters by the use of active nematic gel theory. We characterize flow-alignment coupling, and verify at a quantitative level that compression by flow drives ring formation. Finally, we find that active alignment enhances but is not required for ring formation. Our work characterizes the physical mechanisms of actomyosin ring formation and highlights the role of flow as a central organizer of actomyosin network architecture.

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Background: High directional persistence is often assumed to enhance the efficiency of chemotactic migration. Yet, cells in vivo usually display meandering trajectories with relatively low directional persistence, and the control and function of directional persistence during cell migration in three-dimensional environments are poorly understood. Results: Here, we use mesendoderm progenitors migrating during zebrafish gastrulation as a model system to investigate the control of directional persistence during migration in vivo. We show that progenitor cells alternate persistent run phases with tumble phases that result in cell reorientation. Runs are characterized by the formation of directed actin-rich protrusions and tumbles by enhanced blebbing. Increasing the proportion of actin-rich protrusions or blebs leads to longer or shorter run phases, respectively. Importantly, both reducing and increasing run phases result in larger spatial dispersion of the cells, indicative of reduced migration precision. A physical model quantitatively recapitulating the migratory behavior of mesendoderm progenitors indicates that the ratio of tumbling to run times, and thus the specific degree of directional persistence of migration, are critical for optimizing migration precision. Conclusions: Together, our experiments and model provide mechanistic insight into the control of migration directionality for cells moving in three-dimensional environments that combine different protrusion types, whereby the proportion of blebs to actin-rich protrusions determines the directional persistence and precision of movement by regulating the ratio of tumbling to run times. Keywords: Actin-rich protrusion; Bleb; Cell migration; Directionality; Persistence; Protrusion orientation; Run and tumble.

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We derive the equations for a thin, axisymmetric elastic shell subjected to an internal active stress giving rise to active tension and moments within the shell. We discuss the stability of a cylindrical elastic shell and its response to a localized change in internal active stress. This description is relevant to describe the cellular actomyosin cortex, a thin shell at the cell surface behaving elastically at a short timescale and subjected to active internal forces arising from myosin molecular motor activity. We show that the recent observations of cell deformation following detachment of adherent cells (Maître J-L et al 2012 Science 338 253–6) are well accounted for by this mechanical description. The actin cortex elastic and bending moduli can be obtained from a quantitative analysis of cell shapes observed in these experiments. Our approach thus provides a non-invasive, imaging-based method for the extraction of cellular physical parameters.

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abstract

Segmentation and tracking of cells in long-term time-lapse experiments has emerged as a powerful method to understand how tissue shape changes emerge from the complex choreography of constituent cells. However, methods to store and interrogate the large datasets produced by these experiments are not widely available. Furthermore, recently developed methods for relating tissue shape changes to cell dynamics have not yet been widely applied by biologists because of their technical complexity. We therefore developed a database format that stores cellular connectivity and geometry information of deforming epithelial tissues, and computational tools to interrogate it and perform multi-scale analysis of morphogenesis. We provide tutorials for this computational framework, called TissueMiner, and demonstrate its capabilities by comparing cell and tissue dynamics in vein and inter-vein subregions of the Drosophila pupal wing. These analyses reveal an unexpected role for convergent extension in shaping wing veins.

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We study the effect of turnover of cross-linkers, motors, and filaments on the generation of a contractile stress in a network of filaments connected by passive cross-linkers and subjected to the forces exerted by molecular motors. We perform numerical simulations where filaments are treated as rigid rods and molecular motors move fast compared to the time scale of an exchange of cross-linkers. We show that molecular motors create a contractile stress above a critical number of cross-linkers. When passive cross-linkers are allowed to turn over, the stress exerted by the network vanishes due to the formation of clusters. When both filaments and passive cross-linkers turn over, clustering is prevented and the network reaches a dynamic contractile steady state. A maximum stress is reached for an optimum ratio of the filament and cross-linker turnover rates. Taken together, our work reveals conditions for stress generation by molecular motors in a fluid isotropic network of rearranging filaments.

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Morphological transformations of living cells, such as shape adaptation to external stimuli, blebbing, invagination, or tethering, result from an intricate interplay between the plasma membrane and its underlying cytoskeleton, where molecular motors generate forces. Cellular complexity defies a clear identification of the competing processes that lead to such a rich phenomenology. In a synthetic biology approach, designing a cell-like model assembled from a minimal set of purified building blocks would allow the control of all relevant parameters. We reconstruct actomyosin vesicles in which the coupling of the cytoskeleton to the membrane, the topology of the cytoskeletal network, and the contractile activity can all be precisely controlled and tuned. We demonstrate that tension generation of an encapsulated active actomyosin network suffices for global shape transformation of cell-sized lipid vesicles, which are reminiscent of morphological adaptations in living cells. The observed polymorphism of our cell-like model, such as blebbing, tether extrusion, or faceted shapes, can be qualitatively explained by the protein concentration dependencies and a force balance, taking into account the membrane tension, the density of anchoring points between the membrane and the actin network, and the forces exerted by molecular motors in the actin network. The identification of the physical mechanisms for shape transformations of active cytoskeletal vesicles sets a conceptual and quantitative benchmark for the further exploration of the adaptation mechanisms of cells.

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Although cellular tumor-suppression mechanisms are widely studied, little is known about mechanisms that act at the level of tissues to suppress the occurrence of aberrant cells in epithelia. We find that ectopic expression of transcription factors that specify cell fates causes abnormal epithelial cysts in Drosophila imaginal discs. Cysts do not form cell autonomously but result from the juxtaposition of two cell populations with divergent fates. Juxtaposition of wild-type and aberrantly specified cells induces enrichment of actomyosin at their entire shared interface, both at adherens junctions as well as along basolateral interfaces. Experimental validation of 3D vertex model simulations demonstrates that enhanced interface contractility is sufficient to explain many morphogenetic behaviors, which depend on cell cluster size. These range from cyst formation by intermediate-sized clusters to segregation of large cell populations by formation of smooth boundaries or apical constriction in small groups of cells. In addition, we find that single cells experiencing lateral interface contractility are eliminated from tissues by apoptosis. Cysts, which disrupt epithelial continuity, form when elimination of single, aberrantly specified cells fails and cells proliferate to intermediate cell cluster sizes. Thus, increased interface contractility functions as error correction mechanism eliminating single aberrant cells from tissues, but failure leads to the formation of large, potentially disease-promoting cysts. Our results provide a novel perspective on morphogenetic mechanisms, which arise from cell-fate heterogeneities within tissues and maintain or disrupt epithelial homeostasis.

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How tissue shape emerges from the collective mechanical properties and behavior of individual cells is not understood. We combine experiment and theory to study this problem in the developing wing epithelium of Drosophila. At pupal stages, the wing-hinge contraction contributes to anisotropic tissue flows that reshape the wing blade. Here, we quantitatively account for this wing-blade shape change on the basis of cell divisions, cell rearrangements and cell shape changes. We show that cells both generate and respond to epithelial stresses during this process, and that the nature of this interplay specifies the pattern of junctional network remodeling that changes wing shape. We show that patterned constraints exerted on the tissue by the extracellular matrix are key to force the tissue into the right shape. We present a continuum mechanical model that quantitatively describes the relationship between epithelial stresses and cell dynamics, and how their interplay reshapes the wing.

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Biological tissues must generate forces to shape organs and achieve proper development. Such forces often result from the contraction of an apical acto-myosin meshwork. Here we describe an alternative mechanism for tissue contraction, based on individual cell volume change. We show that during Drosophila dorsal closure (DC), a wound healing-related process, the contraction of the amnioserosa (AS) is associated with a major reduction of the volume of its cells, triggered by caspase activation at the onset of the apoptotic program of AS cells. Cell volume decrease results in a contractile force that promotes tissue shrinkage. Estimating mechanical tensions with laser dissection and using 3D biophysical modeling, we show that the cell volume decrease acts together with the contraction of the actin cable surrounding the tissue to govern DC kinetics. Our study identifies a mechanism by which tissues generate forces and movements by modulating individual cell volume during development.

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Animal cells can sense chemical gradients without moving and are faced with the challenge of migrating towards a target despite noisy information on the target position. Here we discuss optimal search strategies for a chaser that moves by switching between two phases of motion ("run" and "tumble"), reorienting itself towards the target during tumble phases, and performing persistent migration during run phases. We show that the chaser average run time can be adjusted to minimize the target catching time or the spatial dispersion of the chasers. We obtain analytical results for the catching time and for the spatial dispersion in the limits of small and large ratios of run time to tumble time and scaling laws for the optimal run times. Our findings have implications for optimal chemotactic strategies in animal cell migration.

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When cells move using integrin-based focal adhesions, they pull in the direction of motion with large, ∼100 Pa, stresses that contract the substrate. Integrin-mediated adhesions, however, are not required for in vivo confined migration. During focal adhesion-free migration, the transmission of propelling forces, and their magnitude and orientation, are not understood. Here, we combine theory and experiments to investigate the forces involved in adhesion-free migration. Using a non-adherent blebbing cell line as a model, we show that actin cortex flows drive cell movement through nonspecific substrate friction. Strikingly, the forces propelling the cell forward are several orders of magnitude lower than during focal-adhesion-based motility. Moreover, the force distribution in adhesion-free migration is inverted: it acts to expand, rather than contract, the substrate in the direction of motion. This fundamentally different mode of force transmission may have implications for cell-cell and cell-substrate interactions during migration in vivo.

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Single and collective cellular oscillations driven by the actomyosin cytoskeleton have been observed in numerous biological systems. Here, we propose that these oscillations can be accounted for by a generic oscillator model of a material turning over and contracting against an elastic element. As an example, we show that during dorsal closure of the Drosophila embryo, experimentally observed changes in actomyosin concentration and oscillatory cell shape changes can, indeed, be captured by the dynamic equations studied here. We also investigate the collective dynamics of an ensemble of such contractile elements and show that the relative contribution of viscous and friction losses yields different regimes of collective oscillations. Taking into account the diffusion of force-producing molecules between contractile elements, our theoretical framework predicts the appearance of traveling waves, resembling the propagation of actomyosin waves observed during morphogenesis.

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Cell shape is determined by cellular mechanics. Cell deformations in animal cells, such as those required for cell migration, division or epithelial morphogenesis, are largely controlled by changes in mechanical stress and tension at the cell surface. The plasma membrane and the actomyosin cortex control surface mechanics and determine cell surface tension. Tension in the actomyosin cortex primarily arises from myosin-generated stresses and depends strongly on the ultrastructural architecture of the network. Plasma membrane tension is controlled mainly by the surface area of the membrane relative to cell volume and can be modulated by changing membrane composition, shape and the organization of membrane-associated proteins. We review here our current understanding of the control of cortex and membrane tension by molecular processes. We particularly highlight the need for studies that bridge the scales between microscopic events and emergent properties at the cellular level. Finally, we discuss how the mechanical interplay between membrane dynamics and cortex contractility is key to understanding the biomechanical control of cell morphogenesis.

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A cell is a complex material whose mechanical properties are essential for its normal functions. Heating can have a dramatic effect on these mechanical properties, similar to its impact on the dynamics of artificial polymer networks. We investigated such mechanical changes by the use of a microfluidic optical stretcher, which allowed us to probe cell mechanics when the cells were subjected to different heating conditions at different time scales. We find that HL60/S4 myeloid precursor cells become mechanically more compliant and fluid-like when subjected to either a sudden laser-induced temperature increase or prolonged exposure to higher ambient temperature. Above a critical temperature of 52 ± 1°C, we observed active cell contraction, which was strongly correlated with calcium influx through temperature-sensitive transient receptor potential vanilloid 2 (TRPV2) ion channels, followed by a subsequent expansion in cell volume. The change from passive to active cellular response can be effectively described by a mechanical model incorporating both active stress and viscoelastic components. Our work highlights the role of TRPV2 in regulating the thermomechanical response of cells. It also offers insights into how cortical tension and osmotic pressure govern cell mechanics and regulate cell-shape changes in response to heat and mechanical stress.

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Contractile actomyosin rings drive various fundamental morphogenetic processes ranging from cytokinesis to wound healing. Actomyosin rings are generally thought to function by circumferential contraction. Here, we show that the spreading of the enveloping cell layer (EVL) over the yolk cell during zebrafish gastrulation is driven by a contractile actomyosin ring. In contrast to previous suggestions, we find that this ring functions not only by circumferential contraction but also by a flow-friction mechanism. This generates a pulling force through resistance against retrograde actomyosin flow. EVL spreading proceeds normally in situations where circumferential contraction is unproductive, indicating that the flow-friction mechanism is sufficient. Thus, actomyosin rings can function in epithelial morphogenesis through a combination of cable-constriction and flow-friction mechanisms.

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The cortex is a thin, crosslinked actin network lying immediately beneath the plasma membrane of animal cells. Myosin motors exert contractile forces in the meshwork. Because the cortex is attached to the cell membrane, it plays a central role in cell shape control. The proteic constituents of the cortex undergo rapid turnover, making the cortex both mechanically rigid and highly plastic, two properties essential to its function. The cortex has recently attracted increasing attention and its functions in cellular processes such as cytokinesis, cell migration, and embryogenesis are progressively being dissected. In this review, we summarize current knowledge on the structural organization, composition, and mechanics of the actin cortex, focusing on the link between molecular processes and macroscopic physical properties. We also highlight consequences of cortex dysfunction in disease.

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The orderly packing and precise arrangement of epithelial cells is essential to the functioning of many tissues, and refinement of this packing during development is a central theme in animal morphogenesis. The mechanisms that determine epithelial cell shape and position, however, remain incompletely understood. Here, we investigate these mechanisms in a striking example of planar order in a vertebrate epithelium: The periodic, almost crystalline distribution of cone photoreceptors in the adult teleost fish retina. Based on observations of the emergence of photoreceptor packing near the retinal margin, we propose a mathematical model in which ordered columns of cells form as a result of coupling between planar cell polarity (PCP) and anisotropic tissue-scale mechanical stresses. This model recapitulates many observed features of cone photoreceptor organization during retinal growth and regeneration. Consistent with the model's predictions, we report a planar-polarized distribution of Crumbs2a protein in cone photoreceptors in both unperturbed and regenerated tissue. We further show that the pattern perturbations predicted by the model to occur if the imposed stresses become isotropic closely resemble defects in the cone pattern in zebrafish lrp2 mutants, in which intraocular pressure is increased, resulting in altered mechanical stress and ocular enlargement. Evidence of interactions linking PCP, cell shape, and mechanical stresses has recently emerged in a number of systems, several of which show signs of columnar cell packing akin to that described here. Our results may hence have broader relevance for the organization of cells in epithelia. Whereas earlier models have allowed only for unidirectional influences between PCP and cell mechanics, the simple, phenomenological framework that we introduce here can encompass a broad range of bidirectional feedback interactions among planar polarity, shape, and stresses; our model thus represents a conceptual framework that can address many questions of importance to morphogenesis.

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Differential cell adhesion and cortex tension are thought to drive cell sorting by controlling cell-cell contact formation. Here, we show that cell adhesion and cortex tension have different mechanical functions in controlling progenitor cell-cell contact formation and sorting during zebrafish gastrulation. Cortex tension controls cell-cell contact expansion by modulating interfacial tension at the contact. By contrast, adhesion has little direct function in contact expansion, but instead is needed to mechanically couple the cortices of adhering cells at their contacts, allowing cortex tension to control contact expansion. The coupling function of adhesion is mediated by E-cadherin and limited by the mechanical anchoring of E-cadherin to the cortex. Thus, cell adhesion provides the mechanical scaffold for cell cortex tension to drive cell sorting during gastrulation.

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Cytokinesis, the physical separation of daughter cells at the end of mitosis, requires precise regulation of the mechanical properties of the cell periphery. Although studies of cytokinetic mechanics mostly focus on the equatorial constriction ring, a contractile actomyosin cortex is also present at the poles of dividing cells. Whether polar forces influence cytokinetic cell shape and furrow positioning remains an open question. Here we demonstrate that the polar cortex makes cytokinesis inherently unstable. We show that limited asymmetric polar contractions occur during cytokinesis, and that perturbing the polar cortex leads to cell shape oscillations, resulting in furrow displacement and aneuploidy. A theoretical model based on a competition between cortex turnover and contraction dynamics accurately accounts for the oscillations. We further propose that membrane blebs, which commonly form at the poles of dividing cells and whose role in cytokinesis has long been enigmatic, stabilize cell shape by acting as valves releasing cortical contractility. Our findings reveal an inherent instability in the shape of the dividing cell and unveil a novel, spindle-independent mechanism ensuring the stability of cleavage furrow positioning.

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Blebs are spherical membrane protrusions often observed during cell migration, cell spreading, cytokinesis, and apoptosis, both in cultured cells and in vivo. Bleb expansion is thought to be driven by the contractile actomyosin cortex, which generates hydrostatic pressure in the cytoplasm and can thus drive herniations of the plasma membrane. However, the role of cortical tension in bleb formation has not been directly tested, and despite the importance of blebbing, little is known about the mechanisms of bleb growth. In order to explore the link between cortical tension and bleb expansion, we induced bleb formation on cells with different tensions. Blebs were nucleated in a controlled manner by laser ablation of the cortex, mimicking endogenous bleb nucleation. Cortical tension was modified by treatments affecting the level of myosin activity or proteins regulating actin turnover. We show that there is a critical tension below which blebs cannot expand. Above this threshold, the maximal size of a bleb strongly depends on tension, and this dependence can be fitted with a model of the cortex as an active elastic material. Together, our observations and model allow us to relate bleb shape parameters to the underlying cellular mechanics and provide insights as to how bleb formation can be biochemically regulated during cell motility.

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We propose a mechanism for the formation of contractile rings and the apparition of a flow in the cortical layer of cells undergoing cytokinesis at the end of cell division or during the healing of a wound in the cortex of Xenopus eggs. We generalize the hydrodynamic active gel theory along the lines of thin shell theory of continuum elasticity to describe the cell cortex. As in liquid crystal physics, the flow couples to the orientation of the actin filaments. The cortical flow is driven by an increased density of myosin motors in the cortex, and orients the filaments to form the ring.

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The composite and versatile structure of the cytoskeleton confers complex mechanical properties on cells. Actin filaments sustain the cell membrane and their dynamics insure cell shape changes. For example, the lamellipodium moves by actin polymerization, a mechanism that has been studied using simplified experimental systems. Much less is known about the actin cortex, a shell-like structure underneath the membrane that contracts for cell movement. We have designed an experimental system that mimicks the cell cortex by allowing actin polymerization to nucleate and assemble at the inner membrane of a liposome. Actin shell growth can be triggered inside the liposome, which offers a useful system for a controlled study. The observed actin shell thickness and estimated mesh size of the actin structure are in good agreement with cellular data. Such a system paves the way for a thorough characterization of cortical dynamics and mechanics.

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We demonstrate that 3T3 fibroblast cells can exhibit periodic shape oscillations following a loss of cell-substrate adhesion. The oscillatory behavior can last many hours at a constant frequency, and can be switched off and on using chemical agents. We show that the oscillation frequency increases with increasing acto-myosin contractility. The oscillations also cease when extracellular calcium is depleted or when a blocker of calcium channels is introduced. We propose a theoretical description of the oscillations based on an instability of the cortical actin layer. The cortical actin layer is described using the hydrodynamic theory of active gels. We assume that calcium enters the cell via mechanically gated channels and that an increase of the calcium density increases the acto-myosin contractility in the cortical layer. The theory provides a stability diagram for the actin cortical layer showing an oscillatory instability and gives a good description of the oscillation period. We also discuss the connections between these oscillations and other oscillations observed after depolymerization of the microtubules and with the formation of blebs.

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