Cell spreading as a hydrodynamic process

Abstract

Many cell types have the ability to move themselves by crawling on extra-cellular matrices. Although cell motility is governed by actin and myosin filament assembly, the pattern of the movement follows the physical properties of the network ensemble average. The first step of motility, cell spreading on matrix substrates, involves a transition from round cells in suspension to polarized cells on substrates. Here we show that the spreading dynamics on 2D surfaces can be described as a hydrodynamic process. In particular, we show that the transition from isotropic spreading at early time to anisotropic spreading is reminiscent of the fingering instability observed in many spreading fluids. During cell spreading, the main driving force is the polymerization of actin filaments that push the membrane forward. From the equilibrium between the membrane force and the cytoskeleton, we derive a first order expression of the polymerization stress that reproduces the observed behavior. Our model also allows an interpretation of the effects of pharmacological agents altering the polymerization of actin. In particular we describe the influence of Cytochalasin D on the nucleation of the fingering instability.

1 Introduction

Cell motility is a typical example of the difficulty of studying complex systems that exhibit nontrivial small and large-scale dynamics at the same time. From the activity of proteins to changes in shape and cells motion, understanding this phenomenon is a step toward understanding the relationship between local and global behaviors.

In this article, cell spreading is defined as the process during which round cells suspended in a fluid medium become flattened on a 2D solid substrate. Generally, this process does not involve significant motion (< 5μm) of the center of mass of the cell in the plane of the substrate and thus belongs to the class of cell deformations.

At the molecular level, which we will call “microscopic”, the principal player in spreading is the cytoskeletal component actin, which goes from its monomeric G-actin form to its filamentous polymeric and polarized form F-actin. Usually, cell spreading is characterized by the protrusion of a thin layer rich in actin, the lamellipodium. The biochemical reactions of actin filaments have attracted considerable attention both in vitro and in vivo [1, 2, 3]. The main reactions are the growth of filaments by addition of monomers, cross-linking, severing, depolymerization, branching of filaments, and capping of growing filaments. Branching occurs when complexes of actin-related proteins (ARP) 2/3 bind to actin filaments close to the cell membrane. Capping occurs when proteins bind to a filament barbed end (the growing end), preventing further addition of G-actin. Numerous studies have shown the importance of such actin dynamics to motility processes, including the spreading process [1, 2, 3]. The complete set of those biochemical reactions is known to be linked to a complex molecular signaling network which integrates the binding of integrins with the substrate matrix to the actin reactions themselves.

Complimentary to models of the molecular dynamics underlying spreading, recent studies have developed a whole-cell scale point of view, which we call “macroscopic”, that identifies interesting features. The spreading process is seen as being composed of multiple phases, involving different molecular modules. Dobereiner et al. [4] together with Dubin-Thaler et al. [5] and Giannone et al. [6] have systematically identified three “phases” in the spreading process by analyzing the curve of contact area over time. Those phases are illustrated on Figure 1(a–d). Hence, the spreading phases are mathematically-distinguished by having different exponents in their respective power laws for the growth of the cell mean radius versus time. The initial phase (P₀), often called “basal spreading”, involves round cells landing on the substrate and probing the environment for adhesive cues, while keeping a constant radius R = R₀. The following phase (P₁), “fast spreading”, involves a fast growth of the contact area of the cell with the substrate, by protrusion of the lamellipodium, with R ~ t^1/2. Then, the spreading slows down during “late spreading” (P₂), and the cell finally reaches an asymptotic value of contact area. The exponent of this late phase seems to be more variable, with values ranging between 1/10 and 1/4.

Figure 1.

Comparison between driven fluid spreading and cell spreading. (a–d): DIC micrograph of a fibroblast spreading on a fibronectin coated coverslip. Scale bar=10μm. (a) Initial spherical cell in basal spreading P₀. (b) Isotropic protrusion of the cell in fast spreading P₁. (c) Cell at the transition P₁ – P₂ exhibiting the instability nucleation. (d) Cell during late spreading P₂ with well developed fingers. (e–h): Pictures of the spreading of a silicon drop on silicone wafer during spin coating (volume: 50μm, viscosity: 50cm²/s). The substrate is rotating at frequency f = 10Hz. Reprinted figure with permission from F. Melo et al. [10]. Copyright (1989) by the American Physical Society. (i) Scheme of the coordinates system used to describe the spreading in the plane of the substrate.

In hydrodynamic descriptions of fluid spreading in general, power laws that define the evolution of the contact radius are standards outputs. As the exponents of the power laws for cell spreading have been experimentally probed, some researchers have sought to develop hydrodynamic descriptions of cell spreading. In such descriptions, the cell material is described as a continuum, though not necessarily homogeneous. This continuum is then subject to driving and dissipation. For instance, Figure 1(e–h) shows snapshots of a silicon drop spreading under spin coating. In this case, the main driving force, or to be more accurate, stress, is centrifugal, whereas the main dissipation is viscous. Describing cell spreading as a hydrodynamic process therefore involve identifying the relevant driving and dissipation mechanisms. For instance, Cuvelier et al. [7] have proposed a model balancing the viscous dissipation with an “adhesive power”. This model captures the experimental spreading exponents, but in its assumption, the actin is only taken into account as producing a cell cortex in which the viscous dissipation occurs. The model rejects the effect of actin polymerization as a driving force, in contradiction with some experimental data, including those cited above. More recently, Callan-Jones et al. [8] have described the dynamics of cell fragments deformations as a balance between viscous and friction dissipation and the composite activity of an acto-myosin gel. In this case, there is no net driving force and the contact area of the fragment is not increasing in time. As the authors put it, such model “may be appropriate to the study of cells or cell fragments in confined geometries”. Following this trend, we propose a model that adopts an explicit hydrodynamic description of the lamellipodium protrusion during cell spreading. The dissipation is viscous and the driving is what we call “polymerization stresses”. We restrict our “rheology” to the first order expression linking the polymerization stress in the lamellipodium to the deformation of the cell membrane (the main “load” for the actin filaments and the boundary of the system). We assume that the stress generated by the pushing forces due to actin filaments polymerization at the membrane is balanced by the stress generated by the resistance of the membrane. Thus the “active” behavior of cells is hidden in the expression of the stress generated by the pushing forces due to actin filaments dynamics but expressed in term of the resistance of the membrane.

We first detail the original methods we used to keep a rigorous reference frame, necessary to describe spreading in three dimensions over time. We then explain in detail the assumptions of our continuous approach, the continuity equation we use, the momentum balance in the lubrication approximation and the derivation of the polymerization stresses. The third section compares different predictions of our model against experimental data. We derive the spreading law during the fast spreading and we highlight the occurrence of an instability at the leading edge at the transition between the fast and the late spreading phases. As shown on Figure 1, the formation of fingers at the transition P₁/P₂ is reminiscent of a more classical fluid spreading instability [9, 10]. Finally, we discuss the link between our hydrodynamics approach and a more molecular point of view.

2 Material and methods

A framework to describe spreading

Spreading assays

To address the dynamics of cell spreading, we quantitatively record the spreading process of individual fibroblasts by using differential interference contrast microscopy (DIC), with a 20X air objective. Use of fibroblasts is motivated by their model lamellipodial spreading [5]. In this cell type, spreading is essentially achieved by protrusion of a thin cytoskeletal structure, the lamellipodium, which is the focus of our description. In our experimental conditions, fibroblasts rarely exhibit spreading by alternative processes, such as filopodia extension or bleb protrusion. We record the spreading state every 5 s with a CCD camera, producing 16-bit digital gray scale images. We record spreading during a time of up to two hours, depending on the conditions used.

Immortalized mouse embryonic fibroblasts RPTPα+/+ are cultured in Dulbecco’s Modified Eagle’s Medium (DMEM, Gibco) with 10% Fetal bovine serum (FBS, Gibco). One day prior to experiments, cells are sparsely plated to minimize cell-cell interactions prior to suspension. The day of the experiment cells are detached with trypsin/EDTA (0.05% for 1 min), the trypsin is inactivated with soy-bean trypsin inhibitor (1 mg/ml in DMEM), and the cells are suspended in serum free condition in DMEM, and incubated for 30 min before being plated on a coated glass coverslip.

Glass coverslips are acid washed and treated with hexamethyldisilazane, creating a hydrophobic surface that prevents non-specific receptor activation. This surface is coated by 10 μg/ml fibronectin solution for 1h30’ at 37°C.

Cells are treated with various drugs that alter their mechanical and bio-chemical properties. We have analyzed 201 cells in total, across all treatment conditions. Cells are incubated normally for 30 min, but in the presence of the drug, prior to the spreading assay in the same medium. To modify the actin dynamics we use drugs that decrease the polymerization rate, such as Latrunculin A [11] (concentrations: 50nM (number of cells=5), 100nM (n=5), 200nM (n=16) and 300nM (n=6)) and Wiskostatin [12] (concentrations: 5μM (n=8), 10μM (n=8), 20μM (n=6) and 50μM (n=6)), or we use Cytochalasin D [13], which modifies the capping dynamics (concentrations: 50nM (n=6), 100nM (n=10), 200nM (n=6), 500nM (n=5)). To test the influence of myosin we use Blebbistatin [14] (concentrations: 2.5μM (n=4), 5μM (n=3), 50μM (n=6), 100μM (n=7)), Calyculin A [15] (concentrations: 0.2nM (n=6), 1nM (n=7), 5nM (n=10), 20nM (n=4)) and Rho Kinase inhibitor Y-27632 [16] (concentrations: 2μM (n=3), 10μM (n=5), 25μM (n=4), 50μM (n=5)). All the drugs were purchased from Calbiochem, except for the Cytochalasin D, purchased from Sigma.

Recording spreading in the (r,δ) plane

To obtain the geometric parameters of cells in the plane of the substrate, (r, δ) sketched in Figure 1i, we wrote an original fragmentation algorithm to transform the 16-bit digital gray scale images into binary images separating the cell area from the background. This new algorithm allows for the automatic analysis of DIC motility assays, overcoming recurrent difficulties. The algorithm, written in Java, works on the open source software ImageJ (Wayne Rasband, NIH, http://rsb.info.nih.gov/ij/). The separation involves 4 steps: image gradient, threshold, fill holes and noise reduction. The 4 steps are illustrated in Figure 2a. Image gradient uses the ImageJ tool “Find Edges” to take the 2D gradient of the image using the Sobel method. The next step is the automatic selection of a threshold on gray levels to fragment the cell from the background. For a too low threshold, cells are broken into disconnected parts (organelles, nucleus,…). For too high a threshold, the leading edge is not correctly fragmented and its geometry is destroyed by the noise. Solving this problem requires the optimization of the threshold level between those two extremes. The optimization process involves the identification of the critical gray value for the noise percolation transition. The transition is identified by analyzing the pattern of all the clusters of black pixels surrounded by white pixels. This critical gray value is obtained by maximizing the number of clusters in the binary image, or, equivalently, by minimizing the average size of the clusters in the binary image. Figure 2b illustrates the optimization of the threshold level. Once the optimum threshold level is obtained, the image is transformed into a binary image (black cell and white background). The next step uses the ImageJ tool “Fill Holes”. Finally, the resulting small black clusters are removed by keeping only the larger cluster: the cell. Noise on the leading edge is reduced using the ImageJ tool “Despeckle”, which performs a median filter on the image.

Figure 2.

Steps of the fragmentation algorithm. (a) From left to right: original DIC image, 2D gradient found with ‘find edges’, optimum threshold, fill holes, remove noise. (b) Determination of the optimum threshold level at the percolation transition, when the number of clusters is maximum. Inset: output images obtained for different levels of threshold.

The average contact radius is then obtained from the formula $R=\sqrt{A/\pi }$ [7], where A is the contact area. The symmetry of cells is obtained using the circularity defined as C = 4π(A/P²), where P is the cell perimeter. C = 1 corresponds to a circle and the circularity then decreases when the shape becomes more anisotropic. Therefore, the spreading process in the plane of the substrate can be described using a pair of the following quantities: either A or R and either P or C. In the present study we choose the two dimensionless numbers (χ = R/R₀,C) where R₀ is the initial radius of the spherical cell in suspension.

To describe more precisely the fingering instability occurring at the transition P₁/P₂, we construct spatio-temporal diagrams of the spreading, mapping the radius of each point of the leading edge over time. This spatio-temporal map of the radii during spreading is obtained using an algorithm written in Java and executed in ImageJ. We use a cyclic system of coordinates close to the polar coordinates. Each point of the cell edge is identified by its radius ρ to the center of the cell (centroid) and by a cyclic curvilinear coordinate δ along the perimeter. Figure 1i sketches the coordinate system. The leading edge is then composed at each time by a set of points labeled by the pair of coordinates (ρ(t),δ(t)), with δ(t) + P(t) = δ(t), where P(t) is the cell perimeter at time t. A spatio-temporal diagram is then constructed using time and the curvilinear coordinate δ as axes and the radius ρ in color code. Figure 3 gives a example of the basic set of data we extract from the spreading of fibroblasts.

Figure 3.

Typical experimental data set for the first 15 min of spreading of a fibroblast on a fibronectin coated coverslip. (a) DIC micrographs of the cell at different times. Scale bar=10μm. (b) Graph of the reduced radius χ (■) and circularity C (○) vs time (in minutes). (c) Spatio-temporal map of the spreading in the coordinate system (δ, ρ). (d) Graph of ρ = f(δ) at three instants during spreading. The dotted line highlights the beginning of the fingers nucleation (C decreases).

Recording spreading in the (r,h) plane

Even if the main features of spreading geometry exist in the plane of the substrate, the process is three-dimensional. We need to consider the evolution of cell shape in planes perpendicular to the substrate. Such planes can be described by a couple of coordinates (r,h). To obtain the height profile of the cell we use confocal microscopy at different heights from the substrate. Firstly, the spreading is monitored until the cells are fixed by formaldehyde exposure for 15 min. We continue recording to check that fixation does not modify the cell shape. To determine the height profile, we then use confocal fluorescence microscopy of the rhodamine-phalloidin labeled F-actin. The assumption is that the relevant height profile for the model is the height of the actin cytoskeleton. We therefore record the fluorescence of labeled actin at different heights from the substrate, every 50 nm. In this way, we obtain ~20 slices of the actin distribution as a function of the cell height. Such steps are below the absolute optical resolution. Nonetheless, on each slice the intensity is then thresholded (using auto threshold on ImageJ). This step eliminates low fluorescence coming from regions adjacent to the excited one by keeping only the brightest regions. Moreover, it removes artifacts due to variation in the actin concentration at a given height. The threshold always outlines an actin-free circle in the center of the cell. This “hole” is due to the nucleus and we fill it systematically. All the binary slices are then added to create a gray level image reflecting the height of the cell actin cytoskeleton. This step uses the “Z-project” tool of ImageJ.

3 Model

Continuum hydrodynamics

Lubrication approximation and constant volume

In the spreading of liquid droplets, two approximations of the motion of the fluid can lead to a spreading equation [17, 18]. In the framework of cell spreading, we can also adopt the same simplifying assumptions. Namely, due to the high viscosity of cells and to the small velocities and sizes involved, inertia is negligible. Moreover, at the time scales under study, between the time sampling of 5 s and the experiment duration around 15 min, the cell elasticity is weak [19]. This is the first assumption: inertia and elasticity are neglected, in other words, the hydrodynamics is overdamped. The second assumption is linked to the specificity of the spreading geometry. In this geometry, the thickness of the material is much smaller than its radial extent. From this observation, the vertical and horizontal lengths are considered separately. The local stress σ is assumed to depend only on the coordinates parallel to the substrate and the vertical component of the velocity is neglected. These two assumptions allow the simplification of the Navier-Stokes equation (momentum balance) to what is sometime called “lubrication theory” [20].

One immediate consequence of such assumptions is that the radial velocity profile in the vertical direction is parabolic. In the co-moving edge frame, Dussan and Davis have shown [21] that this profile corresponds to a characteristic rolling motion, reminiscent of a caterpillar vehicle. Some fluid elements are going away from the edge, especially near the substrate. Nonetheless, in the laboratory frame, the overall motion corresponds to an outward flow with an average advancing velocity $U=1/h{\int }_0^{h}vdz$. Similarly, during cell spreading, the intracellular material flows out radially even though in the co-moving edge frame, some material can go backward, because of the general rolling motion described above, or for more specific reasons, like the retrograde actin flow associated with the treadmilling mechanism. Since, we only deal with the average edge velocity U, those internal flows are coarse-grained and ignored in the following discussion. Thus, in the lubrication framework, the leading edge velocity U takes a Darcy form, proportional to the stress gradient in the cell:

$$3\eta U=h^2\nabla \sigma$$ (1)

∇ is the 2D differential operator in the (r, δ) plane. η is the viscosity of the cell (obtained through the relaxation of the actin meshwork [22]). Experiments conducted on spreading of Newtonian (η=constant) and non-Newtonian (η = f(σ)) fluids do not detect any fundamental difference in behavior [23]. The spreading law is similar and leading edge instability also occurs. Therefore, in this study, we suppose η independent of the flow velocity. η is the first relevant parameter we identify. It emerges from the coarse-graining of various microscopic dissipation processes that we assume to be ultimately of a viscous type.

In this study, we also use the reasonable assumption that the cell has a constant volume O (no osmotic pressure during the process). The volume conservation leads to a continuity equation:

$$\frac{\partial h}{\partial t}+\nabla (hU)=0$$ (2)

This continuity equation implies Ω = νR²H, where ν includes numerical factors due to the precise shape of the cell. The characteristic height H is the height of the back of the protruding part of the cell (Figure 4a), which we call the “foot” in analogy with polymer melt spreading [18].

Figure 4.

Sketch of the frame transformation to describe the equilibrium between the cytoskeleton and the membrane. (a) Half height profile of the moving cell. (b) Focus on the contact between filaments and the membrane in the co-moving edge frame. (c) Scheme of a line element of membrane. The force exerted by the filaments is balanced by the membrane resistance. The sketch shows the force balance for a filament that depends on the neighboring filaments.

Equilibrium between polymerization pushing force and membrane resistance force

Different studies have emphasized different possible driving mechanisms for the spreading [24, 25, 26, 27, 28, 29], and spreading is probably a combination of several processes. Nevertheless, at least during the fast spreading, the actin polymerization appears to be the dominant term. When polymerizing, the filaments can exert forces on loads. If a set of filaments are polymerizing against a plane surface, typically the cell membrane, the stress normal to the surface (or pressure) can be defined as the ratio of the sum of the filaments forces to the surface area, or in other words, as the ratio of the average force produced by one filament to the average area occupied by one filament at the surface, as illustrated on Figure 4. The normal stress at the surface is given if we know the filament density at the surface and the expression of the force produced by a polymerizing filament.

The theoretical expression of the force produced by polymerization is hardly reachable directly and would depend anyway on parameters difficult to obtain experimentally, like the polymerization rate. Nevertheless, we can obtain this force indirectly by taking the cell membrane as an external constraint acting on the cell cytoskeleton. If the time associated with thermal fluctuations of the membrane is a lot smaller than the characteristic polymerization time, we can make the assumption that the membrane and the cytoskeleton are at equilibrium, not moving with respect to each other. In other words, the growth of the lamellipodium is quasistatic at the edge. With this hypothesis, the force exerted by a filament on the membrane is equal and opposite to the force exerted by the membrane on the filament. This assumption has been used recently in the field [30, 31].

The expression of the membrane resistance force is linked to its deformations. From a statistical viewpoint, the dynamics of a biological membrane are usually described by the Helfrich-Canham energy [32, 33]. This energy possesses two terms, one linked to the membrane tension (scaling as energy/surface) and one linked to the membrane bending rigidity (scaling as energy). When an underlying cytoskeleton is present, membrane tension and bending rigidity need to be renormalized to include an elastic contribution from the cytoskeleton [34]. A statistical description of the fate of a filament pushing a membrane was worked out by Daniels et al. [35]. Depending on the incident angle of filaments, the polymerization rate and the ratios of the elasticity of the filament, the bending rigidity, and surface tension, they found three regimes. The filament can “stall, spiculate, or run away”. A filament running away is moving with respect to the membrane and so it does not follow our assumption of local mechanical equilibrium. In actuality, as the authors remarked, cells avoid this regime, in particular by making sufficiently stiff filaments. In the case relevant to lamellipodium dynamics, we postulate that the situation is intermediate between the stalling and spiculation regimes. Due to the presence of multiple filaments, the filaments do not completely spiculate, which would give rise to filopodia formation, where the deformation of the membrane is high. In contrast, the work to deform the membrane is shared among neighboring filaments and the deformation of the membrane stays locally small.

Instead of a full statistical description of multiple filaments pushing on a membrane, we adopted a rough coarse-graining of the membrane. We postulate that the presence of actin filament ends at the membrane constrains the membrane’s shape to a succession of segments linking the tip of the actin filaments as sketched in Figure 4. In this case, the membrane is really discrete and we suppose that the relevant deformations are only related to the relative positions of all the vertices formed by actin filaments, the deformations of the segments between vertices being integrated out. Hence, in this case, the force generated on the tip of a filament can be derived by the local geometry, defined by the position of a vertex relative to neighboring vertices. In this study, we only use the first neighboring filaments.

In our rough coarse-grained version of the membrane, the length of membrane between two filaments is constant. Nonetheless, at a finer level of description, thermal fluctuations are constantly underplaying and so forces along the membrane line are constantly present. We assume that the magnitude of those forces can be summarized into a parameter γ that we call membrane tension but that is distinct from the “bare” value of the Helfrich-Canham energy. From a rational viewpoint, it is probably a conglomerate of the bare membrane tension, the bare bending energy, the temperature and the cytoskeleton elasticity, most likely a combination close to the membrane “softness” described by Daniels et al. [35]. From an empirical viewpoint, we assume that γ is the typical parameter scaling as force over distance (energy/surface) obtained by pulling tethers of membrane [30]. It is our second relevant parameter. Then, the tangential forces along the membrane have components along the direction of the filaments. For a filament at the membrane, with tangent unit vector n̂ directed outward, the tangential forces along the membrane segments linked to the neighboring filaments can be projected along n̂. The resulting force along the filament reaches:

$$\begin{array}{l}F=\widehat{n}·\gamma (d_1\widehat{n_1}+d_2\widehat{n_2})\\ =-\gamma (d_{1}cos{\alpha }_1+d_{2}cos{\alpha }_2)\end{array}$$ (3)

Where $\widehat{n_1}$ and $\widehat{n_2}$ are the unit vectors going from the tip of the filament to the tip of the neighboring filaments. d₁ and d₂ are the lengths of the membrane segments associated with the filament. Taking the average of all the filament ends at the leading edge, we get:

$$\begin{array}{l}\langle F\rangle =-\gamma \langle d\rangle (cos{\alpha }_1+cos{\alpha }_2)\\ =-2\gamma \langle d\rangle sin\langle \theta \rangle cos\langle \phi \rangle \end{array}$$ (4)

Where 〈δ〉 is the average distance between filaments ends at the membrane. 〈θ〉 is the average angle between the tangent of the membrane at the end of the filament and the associated segment of membrane. 〈φ〉 the average angle between the filament and the normal to the membrane at its end, as sketched on Figure 4c. We then use the equilibrium between the force exerted by the membrane on filaments and the force exerted by the growing filament on the membrane, which is driven by actin polymerization. The final expression of the average normal force exerted by filaments reaches:

$$\langle F_{\perp }\rangle =2\gamma \langle d\rangle sin\langle \theta \rangle {cos}^2\langle \phi \rangle$$ (5)

As discussed above, the normal stress associated with this force is applied to the average area of polyhedrons of membrane delimited by neighboring filaments: s ~ δ². Therefore, the expression of the average normal polymerization stress at the membrane is:

$$\langle {\sigma }_{rr}(R)\rangle =\frac{2\gamma sin\langle \theta \rangle {cos}^2\langle \phi \rangle }{\langle d\rangle }$$ (6)

The effect of a global membrane curvature on the stress in the cell is contained in the term sin 〈θ〉. If we take a one dimensional element of membrane, it could be oriented either along δ or h (or equivalently r using the cell profile h(r)). In the spreading geometry, since R ≫ H, the main curvature is along h. At the first order, when there is only a macroscopic curvature of radius ρ, 〈θ〉 is small and therefore sin〈θ〉 ~ 〈θ〉. Then as sketched on Figure 4b, if we approximate the curved membrane line by a semi circle, we can link its length to the number of vertices n by πρ ~ n 〈δ〉. And we also have n〈θ〉 ~ π, so we reach $\langle \theta \rangle \sim \frac{\langle d\rangle }{\rho }\sim \langle d{\kappa }_0\rangle$, where 〈κ₀〉 is the average global curvature. Generally, the expression of the polymerization stress reaches:

$$\langle {\sigma }_{rr}(R)\rangle =2\gamma {cos}^2\langle \phi \rangle \langle \kappa \rangle$$ (7)

Every part of this expression could eventually vary during the spreading. Nevertheless, in this article, we mainly focus on the effect of curvature. Hence we suppose that both the membrane tension γ and the mean angle 〈φ〉 of the filaments at the membrane are constant during the early spreading. Experiments suggest that 〈φ〉 = ±35° [36]. In the following, we rescale the membrane tension to include this angle, γ cos² 35° → γ.

Curvature expansion

As sketched in Figure 4c, the line element of membrane is linked to curvature through the organization of the vertices. If we suppose that the local deformations of membrane at the scale of the filament spacing are small, i.e. θ is small, we can expand the average curvature in two terms 〈κ〉 = 〈κ₀〉+ε〈κ₁〉, with ε small. 〈κ₀〉 is due to the global curvature of the line element formed by many vertices along h. 〈κ₀〉 is linked to the average angle 〈θ₀〉 at each vertex. In a regular fluid, with molecules much closer to each other, the surface is continuous and this average angle would be null. As discussed above, in the spreading geometry the membrane mainly resists deformation along the vertical direction. The first radius of curvature is H. Thus, the first term in the curvature expansion leads to a convective stress ${\sigma }_0(R)=\frac{2\gamma }{H}$.

〈κ₁〉 is the mean curvature of kinks of membrane. Kinks are strong local deformations of the membrane. In other words, 〈κ₁〉 is linked with the fluctuations of θ. 〈κ₁〉 does not depend on the main curvature but on the concavity of the kink, which along h is associated with ∇²h. Therefore, this term also contains curvature along δ. ε is the fraction of vertices forming kinks, or equivalently, the ratio between the average angle θ and the amplitude of fluctuations of θ. This kink contribution to the curvature leads to a diffusive stress σ₁(R) = 2εγ∇²h. Therefore, the average expression of the normal stress at the membrane is developed using the expansion of the membrane curvature:

$$\begin{array}{l}\langle {\sigma }_{rr}(R)\rangle =2\gamma \langle \kappa \rangle =2\gamma (\langle {\kappa }_0\rangle +\epsilon \langle {\kappa }_1\rangle )\\ =2\gamma (\frac{1}{H}+\epsilon {\nabla }^{2}h)\\ =2\gamma (\frac{\nu R^2}{\mathrm{\Omega }}+\epsilon {\nabla }^{2}h)\end{array}$$ (8)

Where we have linked H to R using the volume conservation. So finally, the polymerization stress at the membrane is found to be proportional to R²(t). Notice that if the geometry is different, corresponding to other motile situations, the curvature can have other dependences on global geometric parameters such as R.

Stress transmission in the lamellipodium

To the first order, equation 8 is the rheological equation of the cell material at the membrane. It relates the stress at the membrane to the deformation of the membrane. To complete this expression, we would need an expression linking the average stress at the membrane 〈σ_rr(R)〉, to the normal stress at any given radius 〈σ_rr(r)〉. In the following derivations, we suppose that the polymerization stress is only present in the lamellipodium, where the actin meshwork is sufficiently dense to transmit stresses. We suppose that the stress is maximum at the edge and decreases to zero at the rear of the lamelliopodium. Therefore, the gradient in the r direction only involves the width w of the lamellipodium, that we suppose constant. It is our third relevant parameter. In this case, the convective stress in the bulk due to the global curvature reaches the value ${\sigma }_0(r)=\frac{2\gamma \nu R^2}{w\mathrm{\Omega }}(r-R)+\frac{2\gamma \nu R^2}{\mathrm{\Omega }}$ for R − w < r < R and zero everywhere else. In comparison, the centrifugal stress in a spinning drop spreading [10] is quadratic in r and present in the whole drop. Thus, in our case, the normal stress gradient reaches:

$$\nabla \sigma =\frac{2\gamma }{w}(\frac{\nu R^2}{\mathrm{\Omega }}+\epsilon {\nabla }^{2}h)$$ (9)

4 Results

Experimental validations of the model predictions

Spreading time

The spreading is driven by a complex polymerization dynamics, but the ensemble average depends only on few macroscopic parameters that produce the characteristic time of spreading. In the previous section, aside from the cell initial radius, we have been able to identify three parameters relevant to the dynamics: the cell viscosity η, the membrane tension γ and the lamellipodium width w. Those three parameters can be combined to form a characteristic time scale of spreading $T_0=\frac{w\eta }{\gamma }$. Taking w ≃ 2μm [6], γ ≃ 10μN/m [7, 38] and η ≃ 500Pa.s [22], we obtain T₀ ≃ 100s. So we get a value consistent with the experimental results, where spreading occurs mostly in the time scale of a few minutes.

Predicting the spreading law

Putting together the continuity equation, the momentum balance and the expression of the polymerization stress leads to a partial differential equation for (r(t), h(t)):

$${\partial }_{t}h+h^2\frac{2\gamma }{w\eta }(\frac{\nu R{(t)}^2}{\mathrm{\Omega }}+\epsilon {\nabla }^{2}h){\partial }_{r}h=0$$ (10)

When we take a smooth leading edge, ε ≪ 1 and therefore we can neglect the diffusive term. The resulting equation has a similar form to the equation of the spreading of liquid down an inclined plane [9, 37]. Supposing a power law dependence for the cell radius, dimensional arguments imply: $R(t)\sim {\mathrm{\Omega }}^{1/3}{(\frac{t}{T_0})}^{\alpha }$. Thus we obtain:

$$\begin{array}{c}{\partial }_{t}h+h^2{(\frac{2\gamma }{w\eta })}^{2\alpha +1}{\mathrm{\Omega }}^{-\frac{1}{3}}{t}^{2\alpha }{\partial }_{r}h=0\\ {\partial }_{t}h+{Bh}^2{t}^{2\alpha }{\partial }_{r}h=0\end{array}$$ (11)

Where B stands for the set of parameters. Numerous relevant solutions to this equation could be found by numeric derivations. Analytically we can obtain the stationary solution at long times by the method of characteristics [37, 10]. Along characteristics of parameter ξ we have the two following equations:

$$\begin{array}{c}\frac{dt}{d\xi }=1\\ \frac{dr}{d\xi }={Bh}^2{t}^{2\alpha }\end{array}$$ (12)

Therefore:

$$\frac{dr}{dt}={Bh}^2{t}^{2\alpha }$$ (13)

Initially, the cell height profile is a function of the initial radius R₀. Putting h = f (R₀) as an initial condition we reach:

$$\begin{array}{c}dr=Bf{(R_0)}^2{t}^{2\alpha }dt\\ \Rightarrow r=R_0+Bf{(R_0)}^2{t}^{2\alpha +1}\end{array}$$ (14)

Therefore, at longer times, when R ≫ R₀, the solution of the spreading equation is independent of the initial conditions and follows:

$$h=B^{\frac{1}{2}}{(R-r)}^{\frac{1}{2}}{t}^{-\frac{2\alpha +1}{2}}$$ (15)

Then, using the volume conservation, we have the scaling $R\sim h^{-1/2}\sim t^{\frac{2\alpha +1}{4}}$, thus $\frac{2\alpha +1}{4}=\alpha$ and finally α = 1/2. Taking T₀ = 100 s, a fit $\chi =a{(\frac{t}{T_0})}^{1/2}+1$ of the spreading data of control cells on fibronectin gives a ≃ 0.3. a ≃ 1 so our scaling seem to be legitimate. Note, however, that the data only extend to less than two decades, which means that the early spreading could as well be fitted by an exponent 1. Such exponent could also be interpreted if we do not keep a constant lamellipodium width w but take w ~ R.

Smoothing stochasticity in transition from P₀ to P₁

The previous section described the stationary spreading at long time. We have seen that the equation (11) admits stationary solutions independent of the initial conditions. This implies the presence of a small characteristic “smoothing time” τ [10]. The memory of initial conditions is lost for time larger than τ, and a stationary regime is reached where h and R evolve according to the scaling obtained by the method of characteristics. Dimensional arguments suggest $\tau =\frac{\eta w}{\gamma }f(\frac{H_0}{{\mathrm{\Omega }}^{1/3}})$, with H₀ the characteristic height of the cell foot. This smoothing time is the time needed to go from a spherical shape with height of the order of Ω^1/3 to a cylindrical shape of characteristic height H₀, which follows the stationary scaling obtained in the previous section. This particular property of the equation of spreading gives sense to the emergence of a global behavior out of a stochastic dynamics of the cytoskeleton. When τ is large enough, the stochasticity of the process can be observed through a short decrease in cell circularity at the transition between P₀ and P₁, the initiation of spreading, followed by a slow increase in circularity at the beginning of P₁. Figure 5 illustrates this stochastic anisotropy experimentally observed for fibroblasts on FN-coated coverslips during the first minutes after spreading initiation.

Figure 5.

Smoothing stochasticity at the transition P₀/P₁ of RPTP α + / + on fibronectin coated coverslips. Cell circularity vs time during the first 5 min of spreading, measured every 2 s. Inset: DIC micrographs at different times. The spreading initiation is first anisotropic due to the inherent stochasticity of the actin dynamics and then smoothed out progressively. Scale bar: 10μm.

Instability nucleation at transition from P₁ to P₂

So far, we have shown that a stationary spreading solution exists, with a circular leading edge extending like t^1/2. Nevertheless, in our experiments, this behavior only extends up to a transition to a different regime (P₂). Besides the identification of this new regime on the change of rate of area growth, the transition between P₁ and P₂ involves a modification of the symmetry of the system from an isotropic edge profile, invariant under rotation, to an anisotropic profile when the cell adopts a typical “flower shape” due to the formation of “fingers” around the edge. As mentioned in the introduction, several situations in liquid spreading predict an instability of a circular leading edge [39, 40, 41]. Fingering instability occurs, for instance, in Newtonian fluids where spreading is driven by gravity [9, 37] or centrifugal forces [10, 41], or in polymer [42] or surfactant solutions [43] where spreading is driven by surface tension gradient. In our case, the driving term comes from the polymerization stresses that we developed above.

The mechanism of the instability described by Brenner [39] is sketched on Figure 6. It is based on the formation of a unstable bump near the edge. The formation of this bump on the height profile is found when adding the term σ₁, depending on the curvature of kinks. This bump on the height profile is then unstable to perturbations with wave vector along the edge. Under such a perturbation, the fluid reacts by flowing in the direction transverse to the main flow. In the troughs of the sinusoidal profile along the edge, the flow goes downhill, to the back of the bumps of the adjacent tips along the edge. Thus, the height of the troughs become smaller than the height of the tips. Since the velocity of the cell material increases like h² due to the term h²∇σ, the tips then move faster than the troughs resulting in the formation of fingers.

Figure 6.

Formation of a bump at the leading edge during the transition P₁/P₂ of the spreading of fibroblasts. Left (i): During the fast spreading P₁, the profile monotonically decreases at the leading edge. Right (ii): At the transition P₁/P₂, a bump of characteristic size l = 2μm forms at the leading edge. (a) DIC micrograph. (b) Confocal slice of the actin meshwork labeled using rhodamine phalloidin, taken at h = 400 ± 25nm. (c) Reconstruction of the cell height profile using the confocal slices (see Methods). Scale bar=10μm. (d) Height profile averaged along the portion of edge marked with a rectangle on (a–c). Before the formation of the bump (left), the red line is the best fit following h ~ r^1/2. Inset: close up of the analyzed edge portion. DIC micrograph at different time before the fixation. Left: Before the formation of the rim, no bump is observed. Right: The bump nucleation is visible through the formation of a bright line near the edge, corresponding to the top of the bump. (e) Instability nucleation mechanism as proposed by Brenner [39].

To generate such secondary dynamics, we would have to take into account the small diffusive stress due to the kink contribution to the curvature. In this case, we can produce a new time scale that corresponds to the characteristic nucleation time of the instability. Dimensional arguments suggest that this characteristic time is $T_c=\frac{\eta w}{\gamma }f(\epsilon )$. In cell spreading, the bump grows in the vicinity of the leading edge, at the end of P₁, after the characteristic time T_c. The characteristic size l of the bump is obtained by balancing the polymerization stress due to kinks curvature with the viscous dissipation, l = H/(3Ca)^1/3 where Ca is our “capillary number” Ca = ηU/γ. This bump on the height profile is clearly visible on the reconstruction made from confocal slices taken during the spreading at the transition between P₁ and P₂ of the cell exhibited in Figure 6. The bump then triggers the instability as explained above, the transverse flow being induced by perturbations of wavelength of the order of l. These perturbations then increase, usually with a constant wavelength, the first growing mode being generally the asymptotic mode. Here again, the characteristic of the instability, i.e nucleation time and size of the bump only depend on macroscopic parameters emerging out of the microscopic dynamics.

Robustness of the instability selected state in P₂

As in classical liquid spreading, the dimensionless critical radius χ_c for the instability onset, as well as the selected mode, i.e, the number of fingers N, are expected to be constant for small ε [10]. To test this robustness we monitored spreading for various conditions. Experimentally, we use a variety of drugs concentrations modifying the cell biochemical composition. We act on the actin dynamics using Wiskostatin [12], Latrunculin A [11] and Cytochalasin D [13]. Wiskostatin inhibits WASP, an upstream regulator of F-actin assembly, therefore decreasing the growing rate. Latrunculin A decreases the pool of G-actin, which also decreases the growing rate. Cytochalasin D binds to the F-actin barbed end, increasing capping. We also test the impact of the actin molecular motor myosin using Calyculin A [15], which inhibits myosin light chain phosphatase and therefore enhance myosin activity, Blebbistatin [14], which inhibits ATPase activity and therefore decreases myosin activity, and Rhokinase inhibitor [16] that also decreases myosin activity by inhibition of ROCK an upstream regulator of myosin. On average, the critical radius χ_c = R_c/R₀ and the number of fingers N are found to be constant in a broad range of conditions. The critical dimensionless radius χ_c is the cell radius that corresponds to the maximum in circularity, at the end of the fast spreading, when the circularity starts to decrease due to finger formation. We find N = 4 ± 1 and χ_c = 1.5 ± 0.1. We first calculate N and χ_c using the control data (i.e without any drug). Then, for each drug in the experiments we compute the average N and χ_c vs the control parameter (drug concentration). If no variation is found in comparison to the control case, those data are added to calculate the total average N and χ_c. Figure 7 gives the plot R_c vs R₀ for the experimental control data. The linear fit gives R_c = χ_cR₀, with χ_c = 1.47 ± 0.02. The number of fingers is obtained by performing a Fourier transform (FT) on the cell orthoradial profile r(δ). The FT gives the wave length λ of the fingers and the number of fingers is computed using N = λ/P, where P is the cell perimeter. The number of fingers and the critical radius stay constant, but drugs have effects on the nucleation time T_c, as shown on Figure 8a. Decreasing polymerization by Latrunculin A or Wiskostatin, or inhibiting myosin by Blebbistatin or Rhokinase inhibitor increase the nucleation time, whereas enhancing myosin activity by Calyculin A decreases the nucleation time. Such effects are expected since those drugs are known to alter the mechanical properties of cells, including cell viscosity η, size of the lamellipodium w and cell membrane tension γ. Nevertheless, the link between the biochemical activity of those drugs and the extent to which the mechanical properties are modified is still an open question that prevents further analysis of those data at this point.

Figure 7.

Plot of the critical radius of control cells vs the initial radius of control cells. The linear fit gives χ_c = 1.47 ± 0.02 (reduced χ² = 0.2).

Figure 8.

Impact of pharmaceutic drugs on the instability nucleation. (a) Robustness of the number of fingers and modification of the nucleation time T_c. (b) Modification of the instability by increasing ε. Graph showing the increase of the number of fingers N in function of the concentration of the drug Cytochalasin D. Error bars are the standard errors taken on the analyzed cells at each concentration. Inset: DIC micrograph showing the characteristic phenotype observed after 20 min of spreading for the control and for cells treated with 500 nM of Cytochalasin D. Scale bar=10μm.

The only sensible effect on the instability is observed for high concentration of Cytochalasin D (Figure 8). The number of fingers increases. Such modification of N is observed in liquid spreading when the lubrication approximation is not really applicable. In cell spreading, the term of importance is the polymerization stress, composed of a convective stress due to the global curvature and of a diffusive stress due to the local curvature of membrane kinks. Balancing these two terms leads to the dimensionless parameter ε that traces back the ratio between the global curvature of the edge κ₀ and the local deformations of curvature κ₁. As discussed in the modeling section, this parameter ε can be seen as a measure of the importance of strong deformation of the membrane. With high concentration of Cytochalasin D, cells develop filopodia that increase the local curvature and therefore ε. In this case, we cannot exclude the sustained formation of spicules as described by D.R. Daniels et al. [35], our expansion of the curvature is not legitimate.

Lack of adhesion leads to a frustrated nucleation

In liquid spreading, the finger formation and shape can be strongly affected by the wetting properties of the substrate [44]. In complete wetting, the equilibrium contact angle is zero, therefore, the tips and the troughs of the fingers can move forward together. In partial wetting, when the equilibrium contact angle is finite, troughs stop protruding as soon as their height decreases enough so that the contact angle is equal to its equilibrium value [39]. For cells, the fast spreading (P₁) has been shown to be independent of the adhesion to the substrate. The only effects of adhesion are on the basal phase (P₀) prior to the spreading initiation, and on the late spreading (P₂). A decrease in concentration of fibronectin on the substrate extend P₀ [5]. Beside this effect on the latent state, the main impact of adhesion is on the state selected by the instability between P₁ and P₂. We believe that the exponent of the spreading law in this regime depend on adhesion. In a similar way to liquid spreading, the adhesiveness of the substrate has an impact on the nucleation of the finger instability. Figure 9 show some situations where the lack of adhesiveness leads to a “frustrated” instability. Figure 9b shows the spreading of a cell on non adhesive substrate. In this situation, the instability nucleation is strongly reduced, the finger lengths are very small. Similar patterns are also obtained in absence of the adhesion protein Talin [45]. Figure 9a shows the spreading of a cell on a circular adhesive patch of fibronectin. When the cell leading edge reaches the boundary of the non adhesive region, the bump seems to form and the instability starts to nucleate, but the formation of the fingers is blocked. In such cases, the bump is even more visible and the transverse flow is observable, since it does not efficiently evacuate the fluid in the troughs. Figure 9a exhibits traveling waves around the edge that are triggered by the transverse flow. Such traveling waves are known to be contained in solutions of the equation of spreading [46]. This mechanism can explain such traveling phenomenon observed in previous studies on cell spreading [47].

Figure 9.

Modification of the instability by frustrating the nucleation by a lack of adhesiveness. (a) Frustrated transition P₁/P₂ observed for spreading on a fibronectin patterned circle (n = 10). Left: DIC micrograph showing a lateral protrusion wave (→). The patterned circle is highlighted in green, the outside region is non-adherent. Right: Radius spatio-temporal map showing the frustration of the nucleation. The perimeter stays roughly constant and the lateral wave is visible through the diagonal shown with the dotted line. The velocity of the wave is v = 4.5μm/min. (b) Frustrated transition P₁/P₂ observed for spreading on nonadherent substrate (n = 3). Right: DIC micrograph showing the analyzed edge portion. Left: Edge portion over time. The edge oscillates, as the nucleation is inefficient. Scale bar=10μm.

5 Discussion

Link with a molecular approach

Microscopic stochastic dynamics of actin

By developing a continuous description we have seen that cell spreading features deterministic dynamics. Our hydrodynamical model describes quantitatively a lamellipodial spreading that depends only on a few macroscopic parameters. In particular we have been able to identify the characteristic time scale of the spreading process: $T_0=\frac{\eta w}{\gamma }$. We derived the spreading law for a circular leading edge and observed that an instability leads to the formation of fingers, in a similar manner to usual fluid spreading. Given the initial radius of a cell, the value of its viscosity and membrane tension, and the size of its lamellipodium, we can predict the time needed to spread and the evolution of the shape of the cell: isotropic at early stages and exhibiting anisotropic finger-like protrusions at later stages.

Nevertheless, the “flow” of the cell material is really due to the polymerization of actin that pushes the membrane forward. These microscopic dynamics driven by the biochemistry of the molecular components of the cytoskeleton are inherently stochastic. The dynamics of the cell leading edge depend on the occurrence and on the proportion of the different filament reactions, growing, severing, branching and capping. The edge velocity would therefore depend on those reaction rates. Thus, the lubrication equation 1, which links stresses in the lamellipodium and edge velocity, suggests a dependence of the reaction rates on forces applied on the filaments. Such force dependent chemistry can be modeled by various models, including end-tracking stepping motors models [48, 49], or Brownian ratchet models [50, 51]. Computations in the framework of those models were usually done on single or independent filaments. Actually, in lamellipodial protrusion, filaments are coupled by the membrane tension, since without any extracellular load, the force on a filament is f_i = p cos(φ)κ(< i >), where the curvature κ(< i >) is a function of the filament neighborhood < i >. Xiong et al. have shown recently that such “interaction” between filaments could generate a collective behavior rationalizing the hydrodynamical approach [52].

Influence of myosin and adhesion

In our hydrodynamics model, we do not take into account adhesion or myosin. Myosin, the molecular motor associated with actin, is known to be critical for the motility of cells. For instance, myosin contraction of the cytoskeleton is involved in the traction forces of the cell that pull up the rear of the cell [56]. Nevertheless, recent studies show that myosin is only involved in the late spreading [56]. Myosin does not affect the spreading law in P₁ and is only needed by the cell to reach a final constant and stable radius.

The precise dynamics to reach this final steady state are still unclear and could involve several processes. In particular, our model cannot predict any robust scaling for the radius growth in P₂. We have seen that a lack of adhesion modifies the selection of a state by the fingering instability. In this regime, we think that a description of the hydrodynamics should include some treatment of the adhesiveness to the substrate. The study by Callan-Jones et al. [8] may be a relevant starting point. Another potential process involved in the selection of the final steady state could be linked to the average angle of the filaments. Since the polymerization stresses involve an effective tension of the membrane including cos² φ, modification of the average angle of filaments at the membrane could trigger the end of spreading. Experimentally, it has been shown recently that the average orientation of actin filaments at the membrane changes between phases of lamellipodium protrusion and pauses [53]. Such general steps involved in motility processes are similar to our phases of fast and late spreading. This study suggests that 〈φ_P1〉 < 〈φ_P2〉. At the transition to the late spreading, the protrusion is then slowed down, the polymerization stress dropping according to cos² φ. Such change in orientation could be trigger by the activity of myosin.

During late spreading, when the protrusion slows down, the membrane resistance remains the same. Therefore, new ways to keep the equilibrium between cytoskeleton and membrane are needed to avoid collapse. In the first place, cells form focal adhesions, but also, they start exocytosis to prevent increase of γ [54]. Finally, since focal adhesions are discrete points at the edge, the unbalanced membrane resistance on the largest part of the leading edge between focal contact must be supported by the cytoskeleton. This “solid” property that allows stress transmission at a large size scale seems to be critically related to the activity of myosin. Such a “solid” property could be obtain in different parameter ranges by the model of active gels describing the rheology of actin-myosin solutions [55].

Therefore, our postulate is that the myosin activity is not involved in the transition from P₁ to P₂, but in maintaining the asymptotic area that cells eventually reach. Absence of myosin does not affect the fast spreading, but results in a collapse of the cell at late stages. The collapse results in the formation of characteristic “C-shape” cells, due to a lost of “cytoplasmic coherence” [56]. Cai et al. also observed that the lack of myosin activity is correlated with a strong reduction of force generation on the substrate. Moreover, even though qualitatively different, cell collapse is also observed in P₂ when cells spread on non adhesive substrates or in the absence of substrate binding proteins (Talin) [45].

6 Conclusion

In 1978 H.P. Greenspan reasonably explained that “an important theoretical objective is the ability to differentiate, at some level of understanding, between cell phenomena that depend mainly on inanimate fluid mechanics, instability for example, and those extraordinary cell processes that characterize life” [57]. Actually, in this article, we show that early cell spreading can be described quantitatively using hydrodynamics driven by polymerization stresses. Cell spreading does not need osmotic pressure or myosin activity. The dynamics follow the lubrication theory, in other words, a version of the universal Navier-Stokes equations adapted to the spreading geometry. Particular details of this specific cell behavior do not come from different dynamics laws but from an original type of stress generated in “living” material. Such stress depends on the global curvature of the membrane. This term, vanishing in classical fluid spreading, remains in cell spreading because stresses are carried by discrete polar filaments. The actin filaments are transmitting the membrane tangential forces to the cell lamellipodium.

The hydrodynamics of the lamellipodium that we have modeled provides a new and clearer description of cell spreading, from the initiation of spreading (transition P₀/P₁) to the formation of fingers around the edge (transition P₁/P₂). We have identify three relevant parameters; the lamellipodium width w, the cell viscosity η, and the membrane tension γ; which all have an empirical significance. A next step would try to link those parameters, relevant at the macroscopic scale to parameters relevant at the microscopic scale. The lamellipodium width would most likely be the output of some modeling of the architecture of the actin meshwork. The cell viscosity would depend on the various dissipation process in the meshwork. And finally, the effective membrane tension would be the output of a statistical description of multiple filaments pushing a membrane, in the spirit of the study by D.R. Daniels et al. [35] or N.S. Gov et al. [58].