Influence of membrane-cortex linkers on the extrusion of membrane tubes

Abstract

The cell membrane is an inhomogeneous system composed of phospholipids, sterols, carbohydrates, and proteins that can be directly attached to underlying cytoskeleton. The protein linkers between the membrane and the cytoskeleton are believed to have a profound effect on the mechanical properties of the cell membrane and its ability to reshape. Here, we investigate the role of membrane-cortex linkers on the extrusion of membrane tubes using computer simulations and experiments. In simulations, we find that the force for tube extrusion has a nonlinear dependence on the density of membrane-cortex attachments: at a range of low and intermediate linker densities, the force is not significantly influenced by the presence of the membrane-cortex attachments and resembles that of the bare membrane. For large concentrations of linkers, however, the force substantially increases compared with the bare membrane. In both cases, the linkers provided membrane tubes with increased stability against coalescence. We then pulled tubes from HEK cells using optical tweezers for varying expression levels of the membrane-cortex attachment protein Ezrin. In line with simulations, we observed that overexpression of Ezrin led to an increased extrusion force, while Ezrin depletion had a negligible effect on the force. Our results shed light on the importance of local protein rearrangements for membrane reshaping at nanoscopic scales.

Introduction

Biological membranes can dynamically change their shape as a response to an external mechanical stress. The application of a localized force to a membrane leads to the formation of membrane tubes or tethers, which are thin, elongated, cylindrical structures (1). Membrane tubes are involved in a wide variety of cellular processes, such as motility, mechanosensing, signaling, and intercellular and intracellular trafficking (2, 3, 4). Tubes can be artificially extracted through hydrodynamic flow experiments, atomic-force microscopy, or optical or magnetic traps, both from reconstituted vesicles and from live cells (5, 6, 7, 8, 9, 10, 11). In the case of giant vesicles and bare membranes, the force-extension curve of the extrusion process has been well characterized both in experiments and in theoretical models (12, 13, 14, 15, 16, 17, 18). The shape of the tube and characteristic extrusion force was used to directly infer the mechanical properties of the bare membrane, where the force was found to depend on the membrane bending rigidity κ and surface tension γ as $F\sim \sqrt{\kappa \gamma }$, and the radius of the tube scaled as $R\sim \sqrt{\kappa /\gamma }$.

Plasma membrane of eukaryotic cells is connected to the underlying cytoskeleton via linking proteins such as Ezrin, Radixin, and Moesin (ERM) and myosin 1 proteins (19,20). The influence of these linkers on the force required to extrude a tube is still a source of discussion. The attachment to the actomyosin skeleton has been suggested to play a role in the extension of membrane tethers by providing resistance to the flow of the membrane toward the tube (21). It was also shown that reducing the attachment of the plasma membrane to the actomyosin cortex via myosin 1b or Ezrin depletion reduces the static force required to extrude a tether in zebrafish progenitor cells (22). A similar result was obtained with myosin 1g in lymphocytes (23). Furthermore, Ezrin phosphorylation that increases membrane-cortex attachment was shown to increase the tether force in lymphoblasts (24).

Taken together, these data have been interpreted as evidence that membrane-cortex attachments effectively change the membrane tension. In this interpretation, the tension is increased by the strength of the membrane-cortex adhesion per unit area, which in turn also increases the force needed to extrude a membrane protrusion (25). However, alternative pictures have also been put forward in which the membrane flows around the membrane-cytoskeleton attachments into the tether, such that the attachments do not contribute to the extrusion force, resulting in a force that resembles that of the bare membrane (26,27).

Here, we use minimal coarse-grained computer simulations to study the role of the cytoskeleton-membrane attachments on the extrusion of membrane tubes in isolation from other effects that might be present in cells. Our simulations do not explicitly assume driving forces but instead only assume the ingredients—a simple model of a fluid membrane that is sparsely attached to the underlying surface. We then carry out computational experiments in which we extrude protrusions and measure the membrane response for a wide range of membrane-cortex attachment densities. We complement our observations with optical-tweezers-based tether pulling on HEK293T cells adhering on a substrate to measure the impact of the ERM protein Ezrin on membrane mechanics.

Our results indicate that the two previously polarized views can in fact coexist: we find that at low membrane-cortex linker densities the extrusion force is not influenced by the cortex attachments and is equal to that of the bare membrane, whereas at high attachment densities, the force strongly increases compared with the bare membrane. At low linker densities, the membrane flows around the linkers, and the linkers avoid entering the tube to minimize the energetic cost of extrusion; hence, the membrane-cortex attachment does not contribute to the force. At high linker densities, however, the linkers cannot avoid entering the tube composition anymore and necessarily start detaching, causing the substantial increase in the extrusion force accompanied by slow tube relaxation dynamics. In agreement with the simulation findings, our experiments show that for cells that display moderate levels of cortex-linker attachments, depleting ERM protein Ezrin has negligible influence on the extrusion force but impedes the tube relaxation, whereas overexpressing Ezrin leads to a significant increase in the force. Our results point to the importance of local effects—the redistribution of the membrane-cortex attachments around the protrusion—when mechanically probing cellular membranes at the nanoscale.

Methods

Simulation details

The membrane was modeled as a dynamically triangulated sheet with hard sphere beads situated at every vertex of a triangular lattice (28, 29, 30). The system contains two types of beads: one type representing the membrane phospholipid patches and another corresponding to ERM proteins. These proteins are responsible for linking the membrane phospholipids and transmembrane proteins to the underlying cytoskeletal components (19). The dynamics of the system are governed by a Monte Carlo algorithm that involves two types of moves: vertex displacement moves and bond flip moves (Fig. S1). The vertex displacement moves mimic the lateral diffusion of lipids and proteins and allow for vertical membrane fluctuations. The bond flip moves ensure that the membrane preserves its fluidity by dynamically rearranging the connectivity. This membrane model captures correct mechanical properties of biological membranes—bending rigidity, fluidity, and the response to external forces. The model, however, does not describe the bilayer explicitly, and the effects, such as interleaflet friction, are not included. The model also does not allow for topology changes, such as poration or tube scission.

A tube is drawn from the membrane by applying an external point force to a predetermined lipid bead. The simulation methodology aims to mimic a slow pulling procedure, such as optical tweezers or an atomic-force microscopy experiment. The system’s energy is calculated using the Hamiltonian described in Eq. 1. This takes into account the energy required for bending a membrane patch, the energy required to increase the surface area, and the harmonic potential imposed to extrude the tube. The membrane-to-cortex attachment energy is controlled by a series of harmonic springs that can unbind at distances sufficiently far from the vertical equilibrium position.

$$\begin{array}{l}\mathcal{H}=\frac{\kappa }{2}\sum _{⟨ij⟩}\left(1-{\mathbf{n}}_i{\mathbf{n}}_j\right)+\gamma A+k_{\text{pulling}}{\left(l-l_0\right)}^2/2+\\ \sum _k{k}_{\text{linking}}{\left(z_i-z_0\right)}^2/2+{\mathcal{H}}_{\text{steric}}\end{array},$$ (1)

where κ is the membrane’s bending rigidity, i and j are pairs of triangles sharing an edge, n_i and n_j are the normal vectors of the two triangles, γ is the membrane surface tension, and A is the total membrane area. k_pulling is the pulling constant that controls the tube extrusion, l is the vertical position of the pulled membrane bead, and l₀ is its target equilibrium position, which is varied between 20σ and 100σ to obtain the force-extension curves. The linkers are attached to the membrane through a harmonic potential controlled by the spring constant k_linking (Fig. 2 a). The vertical position of each linker is denoted by z_i, with z₀ being its equilibrium vertical position. ${\mathcal{H}}_{\text{steric}}$ is a term that accounts for the hard sphere repulsion between beads, forbidding any two beads to be closer than σ. The beads are not allowed to be at distances greater than 3σ to avoid the formation of holes and pores in the membrane.

Figure 2.

Force-extension curve in the presence of membrane linkers. (a) A side view of a tubular structure pulled in the presence of membrane linkers (magenta beads) is shown. The linkers are restrained on the vertical plane with a harmonic potential. (b) Force-extension curves at small and intermediate linker densities are not significantly affected by the presence of linkers. The peak and plateau forces remain constant independent of protein density (legend indicates the protein number density, σ is the simulation unit of length). (c) The force on the tube tip measured for the target extension of 60σ, averaged between 1 × 10⁶ and 2 × 10⁶ Monte Carlo steps. At high expression levels of the proteins, the force increases substantially. Inset shows the fraction of proteins in the composition of the tube. The error bars and shaded regions in (b)–(d) represent the standard error of the measurements over five different simulation seeds. (d) The tube radius decreases as the tube is extruded. The linker density modulates the radius. To see this figure in color, go online.

A single Monte Carlo sweep consists of N_vertices attempts at displacing a vertex and N_vertices /2 bond flip attempts, where N_vertices is equal to the total number of membrane beads. In this case N_vertices = 3600, and the initial vertical side lengths L_x = L_y = 60σ. The acceptance probability of a Monte Carlo move is controlled by a simple Metropolis-Hastings criterion p = min[1,exp(−ΔE/k_BT)], where ΔE is the energy change after a Monte Carlo move, k_B is the Boltzmann constant, and T is the temperature. The simulation is run in the NA_pT ensemble, where N is the total number of particles in the system and A_p is the membrane’s projected area. The bending rigidity is set to κ = 20k_BT, and the surface tension is kept constant at γ = 3k_BT/σ². The pulling constant, k_pulling = 1k_BT/σ², is chosen to ensure that the extrusion is slow enough such that the membrane beads can diffuse and rearrange on the timescale of the tube extrusion. The resulting force matches those in the experiments (∼10 pN) when taking the Monte Carlo unit length to be σ = 10 nm (which also matches the experimentally observed tube radii of tens of nanometers.)

The linkers are subjected to an additional harmonic potential with constant k_linking = 3k_BT/σ², but they are allowed to freely diffuse in the membrane’s plane. The linkers are allowed to unbind if they are displaced from their vertical equilibrium position of z₀ = −3σ by more than 5σ but also to rebind to maintain the simulation’s detailed balance. The choice of the equilibrium position and attachment constant did not significantly affect the results. The membrane is equilibrated for 300,000 Monte Carlo steps, and then the simulations are run for 2 million steps in total. During equilibration, additional box volume change moves are implemented to relax the membrane surface tension. After equilibration, the beads on the membrane’s edge are fixed in place, and the box size is kept fixed during the extrusion. We checked that the system has enough excess membrane area such that the boundary conditions do not affect our results (Fig. S3).

The force profile is recorded by running a set of simulations in which a single randomly chosen central membrane bead is pulled to incrementally increasing equilibrium positions. The membrane will thus exert a restoring force on that bead. The force profile is then reconstructed from the average values of the force at every extension. This procedure allows the reconstruction of the force-extension curve obtained in a static pulling experiment. The force experienced by the protrusion’s tip decreases as the protrusion grows. The force eventually converges to a finite nonzero value as the protrusion reaches its maximal extension (Fig. S2). The measured pulling force is calculated as F_pulling = k_pullingΔz, where k_pulling is the pulling force constant and Δz is the distance between the point being pulled and its vertical equilibrium position. As such, the force will not reach a zero value, and the protrusion will grow until it reaches an extension around which it will fluctuate. We measured the force by averaging it over the course of the 1 million Monte Carlo steps, ensuring that it reached convergence. Each force-extension curve is obtained from averaging over five different random seeds unless otherwise specified.

Results

The force profile for the bare membrane

We first make sure that our model reproduces previously measured force-extension profiles for nonattached membranes. To that end, we apply a vertical pulling force on a small membrane patch in the membrane center. This leads at first to the extrusion of a cone-like structure (Fig. 1 a). The force-extension profile shows a linear elastic response until a critical threshold is reached. The force then drops by around 8%, and the protrusion undergoes a shape transition from the catenoid to a thin tubular structure, as shown in Fig. 1 b, and the extrusion force eventually reaches a plateau. This extension profile is consistent with experimental and theoretical studies of tether extrusion on free membranes that show the familiar profile of a barrier followed by a plateau (12). It has been shown that the height of the barrier is influenced by various factors, such as membrane composition, speed of pulling, and the area of contact between the membrane and the pulling object, whereas the plateau is typically determined only by the membrane bending rigidity and surface tension (14,27,31,32). We next explored how this force-extension curve is influenced by the addition of proteins linking the cytoskeleton to the membrane.

Figure 1.

The membrane is extruded into a tubular structure. (a) The membrane is modeled as a uniform thin sheet (gray beads). The membrane is pulled at a central bead, and a tubular structure is extruded. (b) The force-extension curve for a homogeneous membrane (in the absence of any cortex linkers) shows a peak at an extension of approximately l = 50σ, which corresponds to a transition in the protrusion’s aspect from a truncated cone to a thin elongated tube (σ is the simulation unit length that corresponds to ∼10nm). The curve eventually reaches a plateau, after which the extrusion proceeds at the same force. The shaded area represents the standard error for measurements from 10 different simulation seeds. To see this figure in color, go online.

Extrusion force has a nonlinear dependence on the density of cortex linkers

Cytoskeletal attachments, or linkers, were introduced into the system by connecting a fraction of membrane beads to a virtual plane beneath the membrane by a generic, finite harmonic potential (Fig. 2 a). This bond can be broken if sufficiently extended. Although some membrane-cortex linkers can exhibit more complex behavior, such as catch bond (33), this simple generic form of binding is believed to represent well the ERM proteins (34).

As the membrane tube is extruded from such a pinned membrane, we observe that the linkers tend to disperse away from the tube because there is a high energetic cost associated with opposing the pulling force and entering the tether. This causes the force required for extrusion not to be significantly affected by the linker attachments for low protein fractions, as shown in Fig. 2 b. In the early stages of the tube formation, the force required to reach the same extension is slightly greater at higher protein concentrations (22), but the plateau force does not show a significant difference at low to moderate linker concentrations. However, if the linker density is high enough, the growth of a protrusion requires a large-scale concerted reorganization of the linkers to facilitate the transport of sufficient material into the tubular structure. In this case, the linkers cannot effectively exclude themselves from the tube and will experience a pulling force, at the same time slowing down the dynamics of tube extrusion.

The growth of a protrusion hence becomes expensive, resulting in an exponential increase in the pulling force for higher protein concentrations, as seen in Fig. 2 c, eventually fully preventing tube extrusion. Taken together, our model shows that the extrusion force can have nonlinear dependence on the linker densities: at low densities, the force will not be influenced by the linkers, whereas at higher linker densities, the force will substantially increase.

To make sure that our results are not influenced by the details of our model—namely, the Monte Carlo routine that might not capture the correct dynamics—we repeated our results using a nontriangulated spherical membrane within a molecular dynamics framework. As shown in Figs. S10–S12 and Supporting materials and methods, Section SII H, our results remain qualitatively unchanged; the molecular dynamics model also shows a nonlinear dependence of the force on the linker density (Fig. S12). The only difference is that the critical linker density at which the force increases is shifted to lower values compared with the Monte Carlo model. These results indicate that the nonlinear dependence of the force on the linker density is a robust result, which is not model dependent. However, the point at which the behavior in the force changes from the flat curve to the exponential increase will depend on the details of the model and the system—namely, on the strength of the interactions of the proteins with the cortex and the membrane fluidity.

Finally, we observe that the presence of linkers also causes a slight reduction in the tube radius (Fig. 2 d): as the linker density increases, there is less material freely available for the tube growth, resulting in a decrease of the tube diameter (see also Fig. S11 for the molecular dynamics results).

Linkers are excluded from the tube at intermediate densities

Fig. 3 a further depicts how the linker proteins dynamically rearrange around the tube during tube extrusion in our simulations. To minimize the strain imposed on the membrane, the linkers avoid the region around the tube neck, and also tend to segregate into small protein-rich clusters (Figs. 3 b and S10). The tube itself can contain small concentrations of trapped proteins (Fig. 3 c). We observed that the partitioning of proteins between the tube and the membrane is influenced by the linker binding energy, with higher binding energies resulting in stronger linker exclusion from the tube and smaller tube radii (Figs. S4 and S5).

Figure 3.

The linkers are excluded from the tube at intermediate linker densities. (a) Top view of the membrane before and after the extrusion of the tube. The linkers avoid the tube and often segregate into small clusters. (b) The average separation between the linkers decreases as the tube is extruded from the membrane. The linkers’ cluster size (dark purple line, inset) increases as the extrusion progresses (linkers are considered in the same cluster if the distance between them is less than 3σ). The data are obtained from averaging over 20 simulations in which the tube is extruded to a target length of 1₀ = 60σ. The shading represents the standard error of the measurements over five different simulation seeds. (c) Heatmaps showing the number of linkers per unit of membrane area at low (5%) and high (50%) linker densities. The squares are weighted according to the membrane area they contain. The central region, where the tube is extruded (marked with a red “X” symbol), is depleted of linkers at low densities, showing that proteins prefer to diffuse away from the tube. At high densities, the proteins are distributed uniformly, with a significant fraction trapped inside the tube. To see this figure in color, go online.

Experiments: influence of Ezrin expression on tube extrusion in cells

To qualitatively compare simulation results to experiments, we pulled tethers from adherent cells as depicted in Fig. 4 a with a typical length of 10 μm. As previously shown, the initial force overshoot depends on the interaction area between the bead and the plasma membrane (14) and is not well controlled under experimental conditions (Fig. 4 a, step 3; Fig. S16 a). Upon tube formation, the force next decreases and rapidly reaches a plateau. To measure the static tether force, we waited 10 s to let the force equilibrate (Fig. 4 a, step 4; Fig. S16 a). In agreement with previous reports (35), we confirmed that the tether force does not depend on tether length over a broad range (Fig. S16 b). In addition, special care was taken to ensure the measurement of forces for single tethers only.

Figure 4.

Measuring plateau force for varying Ezrin levels. (a) Force regimes observed during tether pulling from cells corresponding to the steps depicted on the right are shown. A bead at equilibrium position (1) is brought into contact with a cell (2). When the stage is pulled away from the trap, an initial force peak required to detach the membrane from the underlying actin cortex (3) is quickly followed by a stable force F_tether (4). (b) Plateau forces in HEK293T cells overexpressing GFP (n = 19), GFP-EzrinWT (n = 28), or GFP-EzrinTD (n = 21) or treated with control siRNA (sicontrol, n = 36), Ezrin siRNA (siEzrin, n = 23), or ERM siRNA mix (siERM, n = 18) are shown. Mean $\pm$ SD (^∗∗p = 0.008; ^∗p = 0.03). The depletion of Ezrin does not change the tether force compared with the control, whereas the expression of recombinant Ezrin-WT and Ezrin-TD proteins increases the force compared with control. To see this figure in color, go online.

We measured a tether force of 16.0 $\pm$ 2.7 pN in HEK293T cells, which is at the lower end of values reported for adherent cells (Fig. S14; (36)). We checked that the tether force is unchanged when the membrane receptor EphB2 is expressed (HEK-EphB2). It is similar in HeLa cells yet higher in RPE-1 cells (Fig. S14), highlighting that the static tether force varies depending on the cell type. Next, we studied the effect of the manipulation of the expression level of ERM on tether force. Because the transfection process can perturb the cells, we compare the measurements after depletion or overexpression with control cells transfected according to the same protocol but without plasmid. GFP-Ezrin is binding actin in a phosphorylation-dependent manner, whereas the phosphomimetic mutant GFP-EzrinTD permanently links actin to the membrane (Fig. S15 b; (37)).

We found that the depletion of Ezrin or ERM using siRNA (Fig. S15 c; (38)) did not change the tether force compared with the control (Fig. 4 b), whereas the expression of recombinant Ezrin-WT and Ezrin-TD proteins significantly increases the force in HEK cells as compared with the control (Figs. 4 b and S15). These results favorably compare with our simulations, in which we found that the force to extrude tethers increases only when the expression level of linkers is substantially increased. Moreover, depleting Ezrin/ERM enabled faster tube relaxation (Fig. 5 b), in agreement with what has been observed in simulations (Fig. 5 a). These tube relaxation results indicate that the presence of linkers imposes a barrier for the diffusion of membrane components, effectively increasing the local friction (27,39).

Figure 5.

Tether extrusion dynamics depends on the density of linkers. (a) Simulations: as the tube is extruded, the force acting on the protrusion’s tip decays to a constant nonzero value. The dynamics of this relaxation is slower at higher linker density. Each curve represents an average over five simulation repeats. (b) Experiment: upon fast tube extrusion, the force relaxes to a constant nonzero value within a timescale that is faster upon depletion of ERM/Ezrin. To see this figure in color, go online.

Linkers stabilize single isolated membrane tubes

Membrane tubes pulled from bare membranes have been previously shown to merge in each other’s proximity, as the free energy change involved in this process is always favorable (12,40). To test the influence of the linkers on the interactions between protrusions, we go back to our simulations. We pulled two tubes at an initial separation of ΔL = 20σ and allow them to diffuse freely. For bare membranes, we observe that the two tubes grow and eventually merge through the basal truncated cone-like structure into a characteristic Y-shape, as shown in Fig. 6 a. This finding is in qualitative agreement with the structures observed in double tube experiments (40). Interestingly, when the linkers are introduced, the probability of tube fusion decreased substantially and is lower at higher linker densities (Fig. 6 c).

Figure 6.

Protrusion coalescence is hindered by linkers. (a) Tubes coalesce into a Y-shaped structure that transforms into a single long tube. (b) The distance between the pulled points on two tubes decays quickly at low protein densities, showing the tendency of adjacent tubes to coalesce. At higher densities, the linkers will maintain a stable distance, as they are prevented from fusion. The shading represents the standard error of the measurements over 20 different simulation seeds. (c) The percentage of tubes that fuse into a single structure decreases at greater linker densities. To see this figure in color, go online.

To quantify this effect, we measured the relative separation between the tips of the two pulled tubes during the course of the simulation (Fig. 6 b). The average separation between the tubes decreases more slowly at greater protein densities, indicating that the linkers act as obstacles in the merging process. The fusion requires the neck of the tubes to be depleted of linkers before it can take place. The proteins thus act as a barrier for the fusion process, stabilizing isolated tubes. Our findings are in excellent agreement with the study of Nambiar et al. (41), in which the authors found the membrane tethers coalesce more frequently when the membrane-cortex attachment is decreased by depleting myosin 1 proteins, just like in our simulations. It can thus be concluded that the membrane linkers can serve to stabilize protrusive structures involved in physiological processes. Along the same lines, Datar et al. found that the membrane tethers move more easily on the surface of axons when the cytoskeleton is disrupted (39).

Discussion

Using coarse-grained simulations in which lipid components and membrane-cortex attachments are modeled explicitly, we found that the force needed to extrude a protrusion has a nonlinear dependence on the density of the membrane-cortex attachments. At low and moderate densities of the attachments, the amplitude of the force needed to extrude a protrusion is unchanged compared with the bare membrane. In our model, this is caused by 1) the ability of the membrane to flow around the attachments, effectively excluding the attachments from the tube and 2) the ability of the attachments to diffuse away from the tube rather than to oppose the tube extrusion. In contrast, when the linker density is increased above a certain threshold, the force needed for tube extrusion increases significantly, as the linker exclusion out of the tube becomes prohibitively slow, as also demonstrated by a slower relaxation of the force to the equilibrium value.

The linkers’ diffusion in our model effectively captures dynamic ERM attachment/detachment from the cortex (42). To test the influence of the linkers’ diffusion on our key results, we repeated these measurements for the case in which the linkers cannot rearrange and diffuse in the membrane plane but can still detach. As shown in Fig. S9, the nonlinear behavior of the force versus linker density remains unchanged, although the point of crossover was again shifted to lower linkers’ density.

Our simulations indicate that great care should be taken when considering the membrane-cortex attachments in an effective way—for instance, by describing them as a uniform membrane surface adhesion or rescaled membrane tension. Here, we show that the local effects, such as the exclusion of linkers from the tube, can have nontrivial effects on the resulting extrusion force, as well as on the protrusion radius.

Our experimental results align well with the nonlinear effects observed in simulations and show that depletion of Ezrin/ERM does not have significant influence on the plateau force, whereas the overexpression does. Such a response could indicate that the density of linkers in HEK cells might lie close to the transition from a plateau to exponential part of the force versus linker density curve.

Previous studies based on static tether force measurements have reported effects of Ezrin and myosin 1 isoforms on effective membrane tension in nonadherent cells such as lymphocytes (23) or in cells with high blebbing propensity (43). By interfering with the membrane-cortex attachment, Diz-Muñoz et al. showed that a decrease in the attachment led to a decrease in the force required to pull a membrane tether from developing zebrafish embryo (22). Recently, Berget et al. reported that incorporating a synthetic linker that increases the membrane-cortex attachment in embryonic mouse cells led to an increase in the tether extrusion force (44). Similarly, de Belly et al. (45) observed that mouse embryonic stem cells that display membrane blebbing, which is believed to be a sign of low membrane-to-cortex attachment, require lower force for tether extrusion than nonblebbing cells. In all of the above cases, the forces required for tube extrusion from embryonic cells (∼50 pN) were substantially higher than in the case of HEK cells probed here (∼10 pN). It is hence likely that the expression level of membrane-cortex attachments in embryonic cells is naturally higher than in epithelial cells studied here, which would cause the experiments to probe the right-most part of the nonlinear curve presented in Fig. 2 c, where the force severely depends on the linker concentration.

The critical density of linkers after which the force increases is expected to depend on the details of the system as well as on the pulling procedure. It is important to note that, because of the simplicity of our model and the lack of experimental information on the interactions and local densities, it was not our intention to provide quantitative mapping to an exact experimental system. In fact, our simulations clearly show that when using different models that assume different protein-cortex interactions and different degrees of the membrane and linkers’ diffusivities, the value of the critical linker density can substantially change. However, our central result—the nonlinear nature of the force versus linker density curve—remains unchanged.

Our simulations and experiments were carried out in a quasistatic way, which is comparable to the rate of filopodia growth (46) and allows enough time for the linkers to exclude themselves out of the tube for low and moderate linker densities. Such a pulling is however still not infinitely slow, and for high linker densities, the force is not necessarily relaxed. Hence, it can be expected that the critical linker density for which the force increases will be shifted to higher values for even slower pulling and longer equilibration times, both in simulations and in experiments. Likewise, in fast dynamic pulling experiments, the crossover might be shifted to lower linker densities, or the initial plateau of constant force might be completely absent. Interestingly, nonlinear effects similar to those reported here were observed in dynamic tether pulling experiments via hydrodynamic flows in red blood cells (8). The extrusion velocity of the tube was observed to depend on the extrusion force in a nonlinear fashion, which was also attributed to the dynamical feature of membrane-cytoskeleton attachments.

Numerous other effects can be present in live cells that contribute to the membrane-pulling force when the level of ERM proteins is changed. This can, for instance, include changes in cell adhesion and changes in the cytoskeleton architecture and flow, which all might have an influence on the global tension and/or local force. However, even in the minimal system presented in our simulations, it is apparent that the role of membrane-cortex attachments might be nontrivial and that their contribution cannot be necessarily considered as an effective adhesion or an effective tension.

Finally, in simulations, we observed that the linkers have a substantial effect on the stability of protrusions, preventing their diffusion and rendering them more stable against coalescence with other tubes, in line with previous experimental observations of the increased tube coalescence upon depletion of myosin 1 proteins (41). We hope that our simulations and experiments will renew discussions on the mature but still unsolved question of the influence of membrane-cortex attachments on membrane tension and membrane reshaping and stimulate the investigation of local effects and protein dynamics in the formation of membrane protrusions.

Author contributions

A.P. performed and analyzed the Monte Carlo simulations. C.V.C. developed and analyzed the molecular dynamics simulations. A.Š. conceived and supervised the research. T.J.L., E.C., and P.B designed, performed, and analyzed the membrane-pulling experiments. The manuscript was written by A.P., P.B., and A.Š. All authors revised and approved the manuscript.

Editor: Claudia Steinem.