Mechanical instabilities of biological tubes

Plateau-Rayleigh instability of a soft layer coated on a rigid cylinder

We study the Plateau-Rayleigh instability of a soft viscoelastic solid layer coated on a rigid cylinder i.e., a soft fibre with a rigid core. The onset of instability is examined using a linear stability analysis. We find that increasing the rigid cylinder radius or the stiffness of the layer reduces the growth rate of the fastest growing mode. For each rigid cylinder radius, a critical elastocapillary number is found below which all wavelengths of disturbances are stable. The critical value for a soft fibre with a thick rigid cylindrical core can be several orders of magnitudes larger than that for a totally soft fibre (no rigid core). This highlights the strong stabilizing effect of the rigid core on the system. Increasing the relaxation timescale of the viscoelastic material also slows down the growth of disturbances, but has no effect on the critical elastocapillary number. Interestingly, the wavelength of the fastest growing mode is independent of the rigid cylinder radius for the purely elastic case.

Cell division angle predicts the level of tissue mechanics that tune the amount of cerebellar folding

Modeling has led to proposals that the amount of neural tissue folding is set by the level of differential expansion between tissue layers and that the wavelength is set by the thickness of the outer layer. Here, we used inbred mouse strains with distinct amounts of cerebellar folding to investigate these predictions. We identified a distinct critical period during which the folding amount diverges between the two strains. In this period, regional changes in the level of differential expansion between the external granule layer (EGL) and underlying core correlate with the folding amount in each strain. Additionally, the thickness of the EGL varies regionally during the critical period alongside corresponding changes in wavelength. The number of SHH-expressing Purkinje cells predicts the folding amount, but the proliferation rate in the EGL is the same between the strains. However, regional changes in the cell division angle within the EGL predicts both the tangential expansion and the thickness of the EGL. Cell division angle is likely a tunable mechanism whereby both the level of differential expansion along the perimeter and the thickness of the EGL are regionally tuned to set the amount and wavelength of folding.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

Cell division angle regulates the tissue mechanics and tunes the amount of cerebellar folding

Modeling has proposed that the amount of neural tissue folding is set by the level of differential-expansion between tissue layers and that the wavelength is set by the thickness of the outer layer. Here we used inbred mouse strains with distinct amounts of cerebellar folding to investigate these predictions. We identified a critical period where the folding amount diverges between the strains. In this period, regional changes in the level of differential-expansion between the external granule layer (EGL) and underlying core correlate with the folding amount in each strain. Additionally, the thickness of the EGL is regionally adjusted during the critical period alongside corresponding changes in wavelength. While the number of SHH-expressing Purkinje cells predicts the folding amount, the proliferation rate in the EGL is the same between the strains. However, regional changes in the cell division angle within the EGL predicts both the tangential-expansion and thickness of the EGL. Cell division angle is likely a tunable mechanism whereby both the level of differential-expansion and thickness of the EGL are regionally tuned to set the amount and wavelength of folding.

Topological Connection between Vesicles and Nanotubes in Single-Molecule Lipid Membranes Driven by Head-Tail Interactions

Lipid nanotube-vesicle networks are important channels for intercellular communication and transport of matter. Experimentally observed in neighboring mammalian cells but also reproduced in model membrane systems, a broad consensus exists on their formation and stability. Lipid membranes must be composed of at least two molecular components, each stabilizing low (generally a phospholipid) and high curvatures. Strong anisotropy or enhanced conical shape of the second amphiphile is crucial for the formation of nanotunnels. Anisotropic driving forces generally favor nanotube protrusions from vesicles. In this work, we report the unique case of topologically connected nanotubes-vesicles obtained in the absence of directional forces, in single-molecule membranes, composed of an anisotropic bolaform glucolipid, above its melting temperature, T_m. Cryo-TEM and fluorescence confocal microscopy show the interconnection between vesicles and nanotubes in a single-phase region, between 60 and 90 °C under diluted conditions. Solid-state NMR demonstrates that the glucolipid can assume two distinct configurations, head-head and head-tail. These arrangements, seemingly of comparable energy above the T_m, could explain the existence and stability of the topologically connected vesicles and nanotubes, which are generally not observed for classical single-molecule phospholipid-based membranes above their T_m.

A physical model for dynamic assembly of human salivary stem/progenitor microstructures

The in vitro reconstructions of human salivary glands in service of their eventual medical use represent a challenge for tissue engineering. Here, we present a theoretical approach to the dynamical formation of acinar structures from human salivary cells, focusing on observed stick-slip radial expansion as well as possible growth instabilities. Our findings demonstrate the critical importance of basement membrane remodeling in controlling the growth process.

Blood vessel-on-a-chip examines the biomechanics of microvasculature

We use a three-dimensional (3D) microvascular platform to measure the elasticity and membrane permeability of the endothelial cell layer. The microfluidic platform is connected with a pneumatic pressure controller to apply hydrostatic pressure. The deformation is measured by tracking the mean vessel diameter under varying pressures up to 300 Pa. We obtain a value for the Young's modulus of the cell layer in low strain where a linear elastic response is observed and use a hyperelastic model that describes the strain hardening observed at larger strains (pressure). A fluorescent dye is used to track the flow through the cell layer to determine the membrane flow resistance as a function of applied pressure. Finally, we track the 3D positions of cell nuclei while the vessel is pressurized to observe local deformation and correlate inter-cell deformation with the local structure of the cell layer. This approach is able to probe the mechanical properties of blood vessels in vitro and provides a methodology for investigating microvascular related diseases.

Activity-induced instabilities of brain organoids

We present an analytical and numerical investigation of the activity-induced hydrodynamic instabilities in model brain organoids. While several mechanisms have been introduced to explain the experimental observation of surface instabilities in brain organoids, the role of activity has been largely overlooked. Our results show that the active stress generated by the cells can be a, previously overlooked, contributor to the emergence of surface deformations in brain organoids.

Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes

Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localized) instability when one or both of its lateral surfaces are under sufficient surface tension. Phase transition interpretations have also highlighted that the tube admits a final evolved ‘two-phase’ state. How the localized instability initiates and evolves into the final ‘two-phase’ state is still a matter of contention, and this is the focus of the current study. Through a weakly nonlinear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localized bulging or necking depending on whether the axial stretch is greater or less than a certain threshold value. At this threshold value, an exceptionally super-critical kink-wave solution arises in place of localization. A thorough interpretation of the anticipated post-bifurcation behaviour based on our theoretical results is also given, and this is supported by finite-element method simulations.

Modeling and live imaging of mechanical instabilities in the zebrafish aorta during hematopoiesis

All blood cells originate from hematopoietic stem/progenitor cells (HSPCs). HSPCs are formed from endothelial cells (ECs) of the dorsal aorta (DA), via endothelial-to-hematopoietic transition (EHT). The zebrafish is a primary model organism to study the process in vivo. While the role of mechanical stress in controlling gene expression promoting cell differentiation is actively investigated, mechanisms driving shape changes of the DA and individual ECs remain poorly understood. We address this problem by developing a new DA micromechanical model and applying it to experimental data on zebrafish morphogenesis. The model considers the DA as an isotropic tubular membrane subjected to hydrostatic blood pressure and axial stress. The DA evolution is described as a movement in the dimensionless controlling parameters space: normalized hydrostatic pressure and axial stress. We argue that HSPC production is accompanied by two mechanical instabilities arising in the system due to the plane stress in the DA walls and show how a complex interplay between mechanical forces in the system drives the emerging morphological changes.

Plateau-Rayleigh instability in a soft viscoelastic material

A soft cylindrical interface endowed with surface tension can be unstable to wavy undulations. This is known as the Plateau-Rayleigh instability (PRI) and for solids the instability is governed by the competition between elasticity and capillarity. A dynamic stability analysis is performed for the cases of a soft (i) cylinder and (ii) cylindrical cavity assuming the material is viscoelastic with power-law rheology. The governing equations are made time-independent through the Laplace transform from which a solution is constructed using displacement potentials. The dispersion relationships are then derived, which depend upon the dimensionless elastocapillary number, solid Deborah number, and compressibility number, and the static stability limit, critical disturbance, and maximum growth rate are computed. This dynamic analysis recovers previous literature results in the appropriate limits. Elasticity stabilizes and compressibility destabilizes the PRI. For an incompressible material, viscoelasticity does not affect stability but does decrease the growth rate and shift the critical wavenumber to lower values. The critical wavenumber shows a more complex dependence upon compressibility for the cylinder but exhibits a predictable trend for the cylindrical cavity.

Cells into tubes: Molecular and physical principles underlying lumen formation in tubular organs

Tubular networks, such as the vascular and respiratory systems, transport liquids and gases in multicellular organisms. The basic units of these organs are tubes formed by single or multiple cells enclosing a luminal cavity. The formation and maintenance of correctly sized and shaped lumina are fundamental steps in organogenesis and are essential for organismal homeostasis. Therefore, understanding how cells generate, shape and maintain lumina is crucial for understanding normal organogenesis as well as the basis of pathological conditions. Lumen formation involves polarized membrane trafficking, cytoskeletal dynamics, and the influence of intracellular as well as extracellular mechanical forces, such as cortical tension, luminal pressure or blood flow. Various tissue culture and in vivo model systems, ranging from MDCK cell spheroids to tubular organs in worms, flies, fish, and mice, have provided many insights into the molecular and cellular mechanisms underlying lumenogenesis and revealed key factors that regulate the size and shape of cellular tubes. Moreover, the development of new experimental and imaging approaches enabled quantitative analyses of intracellular dynamics and allowed to assess the roles of cellular and tissue mechanics during tubulogenesis. However, how intracellular processes are coordinated and regulated across scales of biological organization to generate properly sized and shaped tubes is only beginning to be understood. Here, we review recent insights into the molecular, cellular and physical mechanisms underlying lumen formation during organogenesis. We discuss how these mechanisms control lumen formation in various model systems, with a special focus on the morphogenesis of tubular organs in Drosophila.

Flow-accelerated platelet biogenesis is due to an elasto-hydrodynamic instability

Blood platelets are formed by fragmentation of long membrane extensions from bone marrow megakaryocytes in the blood flow. Using lattice-Boltzmann/immersed boundary simulations we propose a biological Rayleigh-Plateau instability as the biophysical mechanism behind this fragmentation process. This instability is akin to the surface tension-induced breakup of a liquid jet but is driven by active cortical processes including actomyosin contractility and microtubule sliding. Our fully three-dimensional simulations highlight the crucial role of actomyosin contractility, which is required to trigger the instability, and illustrate how the wavelength of the instability determines the size of the final platelets. The elasto-hydrodynamic origin of the fragmentation explains the strong acceleration of platelet biogenesis in the presence of an external flow, which we observe in agreement with experiments. Our simulations then allow us to disentangle the influence of specific flow conditions: While a homogeneous flow with uniform velocity leads to the strongest acceleration, a shear flow with a linear velocity gradient can cause fusion events of two developing platelet-sized swellings during fragmentation. A fusion event may lead to the release of larger structures which are observable as preplatelets in experiments. Together, our findings strongly indicate a mainly physical origin of fragmentation and regulation of platelet size in flow-accelerated platelet biogenesis.

Computational modeling of active deformable membranes embedded in three-dimensional flows

Active gel theory has recently been very successful in describing biologically active materials such as actin filaments or moving bacteria in temporally fixed and simple geometries such as cubes or spheres. Here we develop a computational algorithm to compute the dynamic evolution of an arbitrarily shaped, deformable thin membrane of active material embedded in a three-dimensional flowing liquid. For this, our algorithm combines active gel theory with the classical theory of thin elastic shells. To compute the actual forces resulting from active stresses, we apply a parabolic fitting procedure to the triangulated membrane surface. Active forces are then dynamically coupled via an immersed-boundary method to the surrounding fluid whose dynamics can be solved by any standard, e.g., Lattice-Boltzmann, flow solver. We validate our algorithm using the Green's functions of Berthoumieux et al. [New J. Phys. 16, 065005 (2014)10.1088/1367-2630/16/6/065005] for an active cylindrical membrane subjected (i) to a locally increased active stress and (ii) to a homogeneous active stress. For the latter scenario, we predict in addition a nonaxisymmetric instability. We highlight the versatility of our method by analyzing the flow field inside an actively deforming cell embedded in external shear flow. Further applications may be cytoplasmic streaming or active membranes in blood flows.

Cerebellar folding is initiated by mechanical constraints on a fluid-like layer without a cellular pre-pattern

Models based in differential expansion of elastic material, axonal constraints, directed growth, or multi-phasic combinations have been proposed to explain brain folding. However, the cellular and physical processes present during folding have not been defined. We used the murine cerebellum to challenge folding models with in vivo data. We show that at folding initiation differential expansion is created by the outer layer of proliferating progenitors expanding faster than the core. However, the stiffness differential, compressive forces, and emergent thickness variations required by elastic material models are not present. We find that folding occurs without an obvious cellular pre-pattern, that the outer layer expansion is uniform and fluid-like, and that the cerebellum is under radial and circumferential constraints. Lastly, we find that a multi-phase model incorporating differential expansion of a fluid outer layer and radial and circumferential constraints approximates the in vivo shape evolution observed during initiation of cerebellar folding.

Biophysics in oviduct: Planar cell polarity, cilia, epithelial fold and tube morphogenesis, egg dynamics

Organs and tissues in multi-cellular organisms exhibit various morphologies. Tubular organs have multi-scale morphological features which are closely related to their functions. Here we discuss morphogenesis and the mechanical functions of the vertebrate oviduct in the female reproductive tract, also known as the fallopian tube. The oviduct functions to convey eggs from the ovary to the uterus. In the luminal side of the oviduct, the epithelium forms multiple folds (or ridges) well-aligned along the longitudinal direction of the tube. In the epithelial cells, cilia are formed orienting toward the downstream of the oviduct. The cilia and the folds are supposed to be involved in egg transportation. Planar cell polarity (PCP) is developed in the epithelium, and the disruption of the Celsr1 gene, a PCP related-gene, causes randomization of both cilia and fold orientations, discontinuity of the tube, inefficient egg transportation, and infertility. In this review article, we briefly introduce various biophysical and biomechanical issues in the oviduct, including physical mechanisms of formation of PCP and organized cilia orientation, epithelial cell shape regulation, fold pattern formation generated by mechanical buckling, tubulogenesis, and egg transportation regulated by fluid flow. We also mention about possible roles of the oviducts in egg shape formation and embryogenesis, sinuous patterns of tubes, and fold and tube patterns observed in other tubular organs such as the gut, airways, etc.

Role of extrinsic mechanical force in the development of the RA-I tactile mechanoreceptor

Rapidly adapting type I (RA-I) mechanoreceptors play an important role in sensing the low-frequency vibration aspects of touch. The structure of the RA-I mechanoreceptor is extremely complex regardless of its small size, limiting our understanding of its mechanotransduction. As a result of the emergence of bioengineering, we previously proposed an in vitro bioengineering approach for RA-I receptors to overcome this limitation. Currently, the in vitro bioengineering approach for the RA-I receptor is not realizable given the lack of knowledge of its morphogenesis. This paper demonstrates our first attempt to interpret the cellular morphogenesis of the RA-I receptor. We found indications of extrinsic mechanical force nearby the RA-I receptor in the developing fingertip. Using a mechanical compression device, the axon of dorsal root ganglion (DRG) neurons buckled in vitro into a profile that resembled the morphology of the RA-I receptor. This work encourages further implementation of this bioengineering approach in tactile receptor-related research.

Human Brain Organoids on a Chip Reveal the Physics of Folding

Human brain wrinkling has been implicated in neurodevelopmental disorders and yet its origins remain unknown. Polymer gel models suggest that wrinkling emerges spontaneously due to compression forces arising during differential swelling, but these ideas have not been tested in a living system. Here, we report the appearance of surface wrinkles during the in vitro development and self-organization of human brain organoids in a micro-fabricated compartment that supports in situ imaging over a timescale of weeks. We observe the emergence of convolutions at a critical cell density and maximal nuclear strain, which are indicative of a mechanical instability. We identify two opposing forces contributing to differential growth: cytoskeletal contraction at the organoid core and cell-cycle-dependent nuclear expansion at the organoid perimeter. The wrinkling wavelength exhibits linear scaling with tissue thickness, consistent with balanced bending and stretching energies. Lissencephalic (smooth brain) organoids display reduced convolutions, modified scaling and a reduced elastic modulus. Although the mechanism here does not include the neuronal migration seen in in vivo, it models the physics of the folding brain remarkably well. Our on-chip approach offers a means for studying the emergent properties of organoid development, with implications for the embryonic human brain.

Mechanical Regulation of Three-Dimensional Epithelial Fold Pattern Formation in the Mouse Oviduct

Epithelia exhibit various three-dimensional morphologies linked to organ function in animals. However, the mechanisms of three-dimensional morphogenesis remain elusive. The luminal epithelium of the mouse oviduct forms well-aligned straight folds along the longitudinal direction of the tubes. Disruption of the Celsr1 gene, a planar cell polarity-related gene, causes ectopically branched folds. Here, we evaluated the mechanical contributions of the epithelium to the fold pattern formation. In the mutant oviduct, the epithelium was more intricate along the longitudinal direction than in the wild-type, suggesting a higher ratio of the longitudinal length of the epithelial layer to that of the surrounding smooth muscle (SM) layer (L-Epi/SM ratio). Our mathematical modeling and computational simulations suggested that the L-Epi/SM ratio could explain the differences in fold branching between the two genotypes. Longitudinal epithelial tensions were increased in well-aligned folds compared with those in disorganized folds both in the simulations and in experimental estimations. Artificially increasing the epithelial tensions suppressed the branching in simulations, suggesting that the epithelial tensions can regulate fold patterning. The epithelial tensions could be explained by the combination of line tensions along the epithelial cell-cell boundaries with the polarized cell arrays observed in vivo. These results suggest that the fold pattern is associated with the polarized cell array through the longitudinal epithelial tension. Further simulations indicated that the L-Epi/SM ratio could contribute to fold pattern diversity, suggesting that the L-Epi/SM ratio is a critical parameter in the fold patterning in tubular organs.

Finite-wavelength surface-tension-driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid

We deploy linear stability analysis to find the threshold wavelength (λ) and surface tension (γ) of Rayleigh-Plateau type "peristaltic" instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius R_{0}. First we consider a solid cylinder, and recover the well-known, infinite-wavelength instability for γ≥6μR_{0}, where μ is the solid's shear modulus. Second, we consider a volume-conserving (e.g., fluid filled and sealed) cylindrical cavity through an infinite solid, and demonstrate infinite-wavelength instability for γ≥2μR_{0}. Third, we consider a solid cylinder embedded in a different infinite solid, and find a finite-wavelength instability with λ∝R_{0}, at surface tension γ∝μR_{0}, where the constants depend on the two solids' modulus ratio. Finally, we consider an empty cylindrical channel (or filled with expellable fluid) through an infinite solid, and find an instability with finite wavelength, λ≈2R_{0}, for γ≥2.543...μR_{0}. Using finite-strain numerics, we show such a channel jumps at instability to a highly peristaltic state, likely precipitating it's blockage or failure. We argue that finite wavelengths are generic for elastocapillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.

Dendrite fragmentation in alloy solidification due to sidearm pinch-off

Dendrite sidebranch detachment is an important fragmentation mechanism during the solidification of alloys. The detachment occurs at the junction between a sidearm and its parent stem. While this pinching process is driven by capillarity, the presence of solidification opposes the instability. Using a simple numerical model of a single sidearm, we are able to capture the essential dynamics of dendrite sidebranch development and the resulting morphological transitions. While shortly before pinch-off the neck itself obeys well-known universal scaling relations, the overall evolution of the sidearm shape sensitively depends on its initial geometry and the rate of solidification. It is found that pinch-off only occurs over limited ranges of geometrical parameters and cooling rates and is generally bounded by sidearm retraction and coalescence regimes. Simple scaling relations are identified that provide the bounds for the pinch-off regime. Pinching at the branching point is shown to be faster than the Rayleigh-Plateau instability of an infinitely long cylinder.

Complex instability of axially compressed tubular lipid membrane with controlled spontaneous curvature

Tubular lipid membranes (TLMs) are formed by an external pulling force from artificial or biological bilayer vesicles and can be subsequently stabilized by incorporating proteins or amphiphilic polymers into the lipid bilayer. The arising spontaneous curvature of the lipid sheet allows switching off the pulling force without TLM destabilization. However, here we show that during this process two different thermal fluctuation modes drastically increase their amplitudes making fluctuations of the TLM much greater than its radius. Due to the system's proximity to the critical fluctuation point, a weak axial compressive force is sufficient to destabilize the TLM. Its absolute value is shown to be much smaller than that of the pulling force required for the initial lipid nanotube formation. Induced complex instability was studied in the frame of Landau phase transition theory. The process involves two consecutive second-order phase transitions and leads to the tube deformation combining annular corrugation with completely unconventional chiral buckling.

Elastocapillary Instability in Mitochondrial Fission

Mitochondria are dynamic cell organelles that constantly undergo fission and fusion events. These dynamical processes, which tightly regulate mitochondrial morphology, are essential for cell physiology. Here we propose an elastocapillary mechanical instability as a mechanism for mitochondrial fission. We experimentally induce mitochondrial fission by rupturing the cell's plasma membrane. We present a stability analysis that successfully explains the observed fission wavelength and the role of mitochondrial morphology in the occurrence of fission events. Our results show that the laws of fluid mechanics can describe mitochondrial morphology and dynamics.

Balance between apical membrane growth and luminal matrix resistance determines epithelial tubule shape

The morphological stability of biological tubes is crucial for the efficient circulation of fluids and gases. Failure of this stability causes irregularly shaped tubes found in multiple pathological conditions. Here, we report that Drosophila mutants of the ESCRT III component Shrub/Vps32 exhibit a strikingly elongated sinusoidal tube phenotype. This is caused by excessive apical membrane synthesis accompanied by the ectopic accumulation and overactivation of Crumbs in swollen endosomes. Furthermore, we demonstrate that the apical extracellular matrix (aECM) of the tracheal tube is a viscoelastic material coupled with the apical membrane. We present a simple mechanical model in which aECM elasticity, apical membrane growth, and their interaction are three vital parameters determining the stability of biological tubes. Our findings demonstrate a mechanical role for the extracellular matrix and suggest that the interaction of the apical membrane and an elastic aECM determines the final morphology of biological tubes independent of cell shape.

Growth, homeostatic regulation and stem cell dynamics in tissues

The regulation of cell growth in animal tissues is a question of critical importance: most tissues contain different types of cells in interconversion and the fraction of each type has to be controlled in a precise way, by mechanisms that remain unclear. Here, we provide a theoretical framework for the homeostasis of stem-cell-containing epithelial tissues using mechanical equations, which describe the size of the tissue and kinetic equations, which describe the interconversions of the cell populations. We show that several features, such as the evolution of stem cell fractions during intestinal development, the shape of a developing intestinal wall, as well as the increase in the proliferative compartment in cancer initiation, can be studied and understood from generic modelling which does not rely on a particular regulatory mechanism. Finally, inspired by recent experiments, we propose a model where cell division rates are regulated by the mechanical stresses in the epithelial sheet. We show that pressure-controlled growth can, in addition to the previous features, also explain with few parameters the formation of stem cell compartments as well as the morphologies observed when a colonic crypt becomes cancerous. We also discuss optimal strategies of wound healing, in connection with experiments on the cornea.

Forces in tissue morphogenesis and patterning

During development, mechanical forces cause changes in size, shape, number, position, and gene expression of cells. They are therefore integral to any morphogenetic processes. Force generation by actin-myosin networks and force transmission through adhesive complexes are two self-organizing phenomena driving tissue morphogenesis. Coordination and integration of forces by long-range force transmission and mechanosensing of cells within tissues produce large-scale tissue shape changes. Extrinsic mechanical forces also control tissue patterning by modulating cell fate specification and differentiation. Thus, the interplay between tissue mechanics and biochemical signaling orchestrates tissue morphogenesis and patterning in development.

Islands of conformational stability for filopodia

Filopodia are long, thin protrusions formed when bundles of fibers grow outwardly from a cell surface while remaining closed in a membrane tube. We study the subtle issue of the mechanical stability of such filopodia and how this depends on the deformation of the membrane that arises when the fiber bundle adopts a helical configuration. We calculate the ground state conformation of such filopodia, taking into account the steric interaction between the membrane and the enclosed semiflexible fiber bundle. For typical filopodia we find that a minimum number of fibers is required for filopodium stability. Our calculation elucidates how experimentally observed filopodia can obviate the classical Euler buckling condition and remain stable up to several tens of . We briefly discuss how experimental observation of the results obtained in this work for the helical-like deformations of enclosing membrane tubes in filopodia could possibly be observed in the acrosomal reactions of the sea cucumber Thyone, and the horseshoe crab Limulus. Any realistic future theories for filopodium stability are likely to rely on an accurate treatment of such steric effects, as analysed in this work.