Hydrodynamic narrowing of tubes extruded from cells

Abstract

We discuss the pulling force f required to extrude a lipid tube from a living cell as a function of the extrusion velocity L̇. The main feature is membrane friction on the cytoskeleton. As recently observed for neutrophils, the tether force exhibits a “shear thinning” response over a large range of pulling velocities, which was previously interpreted by assuming viscoelastic flows of the sliding membrane. Here, we propose an alternative explanation based on purely Newtonian flow: The diameter of the tether decreases concomitantly with the increase of the membrane tension in the lipid tube. The pulling force is found to vary as L̇^{$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$}, which is consistent with reported experimental data for various types of cells.

Many cellular processes [such as intracellular trafficking (1) or intercellular organelle transport (2)] involve the formation of thin tubular structures known as tethers. Membranous tails also are observed to be left by migrating cells in culture dishes (3). Tethers can be extracted from synthetic vesicles or living cells by the application of an external point force [using a fluid drag (4, 5) or pipette-tweezer system (6, 7)]. In the case of living cells, where the lipidic membrane is coupled to a cytoskeleton, tethers can be used as membrane sensors to measure the membrane–cytoskeleton adhesion energy W₀ (8–12).

Our aim here is to describe the formation of a tether and to derive the required pulling force as a function of the extrusion velocity. Pulling a tube from a cell membrane implies a surface flow of lipid from the cell body to the tether through the membrane–cytoskeleton binders. This viscous drag gives rise to an increase of the membrane tension in the tube and a decrease of its radius.

Statics of Extrusion

We follow the thermodynamic analysis of tether formation proposed by Waugh and coworkers (7, 13, 14), Evans and Yeung (15), and Derényi and coworkers (16) for lipidic bilayers and extended to cell membranes by Sheetz and coworkers (10, 12). As shown in Fig. 1, the cell is usually held by micropipette suction (pressure −ΔP) that sets the membrane tension of the cell σ: 2σ ≃ R_pΔP, where R_p is the radius of the micropipette. The length of the tongue in the pipette is L_p. Another case of experimental interest is found when cells are spread onto an adhesive surface, which ensures that the membrane tension remains weak or constant during the time course of the experiment. The tube is then either extruded by pulling out a small bead adhering to the membrane via micromanipulation if the cell is firmly adhered or, more simply, by applying a hydrodynamic flow over the cell in case of a discrete and sparse adhesion site. The length of the tube is L, the membrane tension of the tube is σ_t, and its radius is r_t. We want to relate r_t and σ_t to σ, the tension of the cell body, and to W₀, the adhesion energy of the membrane to the cytoskeleton. The pulling force f₀ is deduced from the following four equations.

1. The distribution of areas:

   
2. The Laplace law:

   
3. The work of extrusion:

   

   where the first term describes the curvature energy (κ is the curvature modulus), the second term is the work against suction, and the last term describes membrane–cytoskeleton separation. Eqs. 1, 2, and 3 lead to

   

   with σ_t = σ + W₀. The tension of the tube membrane σ_t is higher than the tension of the bound membrane σ.
4. The cell/tube coexistence: The hydrostatic pressures in the two compartments have to be equal.

   

   where R is the radius of the cell. This leads to the classical expression for the tube radius (10, 14, 15):

   

   Finally, from Eqs. 4 and 6, we find

   

   Thus, measuring the static force f₀ yields a direct estimate of W₀.

   Note that Eq. 7 was previously derived by Hochmuth et al. (10) in a different manner. Our derivation allows us to set the theoretical framework used in the next sections for the dynamics of extrusion.

Fig. 1.

Paradigmatic cases of tether extrusion. (a) The cell (radius R) is held in a micropipette (radius R_p). The aspiration pressure ΔP sets the membrane tension σ. The aspirated tongue (length L_p) serves as a lipid reservoir. (b) The cell is spread onto a surface. (c) Adhesion sites between cell and surface are restricted to points. In b and c, the membrane tension is supposed to be weak or to remain constant during the time course of the experiment. Tethers (length L, radius r_t) are extruded by applying a punctual force f either by micromanipulation (a and b) or by hydrodynamic flow (c). Unbinding of the membrane from the cytoskeleton leads to σ_t = σ + W₀ in quasistatic conditions (W₀ is the adhesion energy between membrane and cytoskeleton).

Dynamics of Extrusion

Let us consider a lipidic membrane in which integral proteins are embedded and anchored to the underlying cortical cytoskeleton with a surface density ν ∼ 1/ξ², where ξ is the average distance between bonds. We now extrude the tether at a velocity L̇. We may expect three different regimes.

1. At ultra-low extrusion velocities L̇ < V₁, where V₁ = V₀ exp(−B/k_BT) is the spontaneous dissociation velocity of the binders, the membrane composed of lipids and integral proteins behaves like an ultra-viscous sheet. By assuming that B ∼ 20k_BT is a standard activation energy of the bond and that V₀ ∼ 10 m/s is a typical thermal velocity, we obtain V₁ ∼ 0.01 μm/s. This ultra-slow viscous regime is discussed in Appendix.

2. At intermediate velocities, V₁ < L̇ < L̇_c, the proteins remain bound to the cytoskeleton and the lipids flow around. The friction force on the binders gives rise to a dynamic increase of membrane tension, which corresponds to the permeation regime. The value of L̇_c is given in Discussion of L̇_c.

3. At ultra-high velocities, L̇ > L̇_c, the friction force on the binders that increases linearly in L̇ overcomes the tear-out force (∼ln L̇). In this limit, the binders are torn out, leading to the slippage of the membrane on the cytoskeleton.

Here, we shall focus on the permeation regime, which corresponds to the experimentally explored range of extrusion velocities. The viscous dissipation is generated by the surface flow of lipids toward the tube through the cytoskeleton network. The velocity of the bilayer has to be canceled on each binding protein. If ṙ is the velocity at a radial distance r from the tube axis, the associated viscous dissipation per binder is the product of the bilayer viscosity η_b, the velocity gradient squared (ṙ/ξ)², and the volume of dissipation ξ²e, where e is the thickness of the bilayer.

This bilayer flow induces backflows inside and outside the cell. The viscous losses are dominated by the flow inside the cell, which is confined in a thin layer of thickness h. The dissipation per binder is the product of the cytosol viscosity η_i, the velocity gradient squared (ṙ/h)², and the volume of dissipation ξ²h.

The total viscous dissipation per unit area is balanced by a mechanical restoring force, leading to

with the surface viscosity η_e = k₁η_be + k₂η_iξ²/h (Pa·s·m), where k₁ is a geometrical factor (17, 18) (k₁ = 4π/ln(ξ/b) ∼ 1 − 10with b the radius of an integral protein) and k₂ξ²/h is an effective slippage thickness. In general, this effective thickness may be much larger than k₁e, so that, even if η_i is smaller than η_b, both contributions may be equally important.

Lipid conservation imposes the equality of lipids currents J (Fig. 2):

Fig. 2.

Sketch of the lipid flow from cell body to tether. Distance and velocity scales are indicated (see text for details).

The balance of Eq. 8 leads to the membrane tension gradient

The tension σ_t of the tube is

In the steady-state regime, we assume that the tube achieves a local equilibrium so that Eq. 6 remains valid and rewrites as

The force on the tube is then

with σ_t given by Eq. 11 and r_t by Eq. 12.

Limit of High Friction.

When viscous friction dominates, one readily gets

which leads to

Thus, f(L̇) is not linear in velocity, although we assumed a Newtonian fluid. Nonlinear dependence of force on velocity (with an exponent <1) is often the signature of shear thinning for viscoelastic melts. Our interpretation is different: When the pulling becomes faster, the tension σ_t of the tube increases and its radius decreases. The force is not a simple sum of static plus friction force. As we will show, our proposed model allows a satisfactory reinterpretation of previously reported data on tethers extracted from various cells.

General Expression.

Eqs. 11, 12, and 13 combine as

where f₀ is the static extrusion force (Eq. 7).

Note that, because R ≫ r_t, ln(R/r_t) can be considered as a constant of value a few unities.

Discussion of L̇_c.

At a distance r from the tube axis, the friction force per binder due to the flow of lipids, f_v = η_eṙ, increases linearly with ṙ. As shown by Evans (19), the tear-out force for an individual bond increases logarithmically with ṙ and is given by φ = (k_BT/a) ln (ṙ/V₁), where a ∼ 1 nm is the maximal bond length beyond which the complex dissociates and V₁, the spontaneous dissociation velocity for φ equals zero. The equality between the viscous force f_v and the tear-out force φ defines the crossover velocity L̇_c above which the bond dissociates. The threshold force φ_c is then given by

For example, in the case of neutrophils, one finds η_e ∼ 10⁻⁶ Pa·s·m (1 pN·s/μm). This leads to L̇_c ∼ 100 μm/s.

Proposed Reinterpretation of Experimental Results

Recently, Heinrich et al. (20) experimentally investigated the force–velocity curve for tethers extracted from neutrophils by using a biomembrane force probe. Quite surprising, the plateau force exhibited a nonlinear dependence on pulling speed as f ∼ L̇^0.25. By contrast, earlier works performed on different and often more limited ranges of pulling velocities had revealed a linear force–velocity relation, in agreement with the prevalent model (10): f = f₀ + 2πη_effL̇ (with η_eff the effective membrane viscosity). Evans and coworkers (20) resolved this apparent discrepancy by suggesting a shear-thinning model for tether flow. Thus, they demonstrated that all data were consistent. Although attractive, the proposed model remains purely phenomenological and cannot account for a residual force at zero velocity. Here, we wish to show that our physical model, which ignores any presumed viscoelastic effect but simply takes into account the force dependence of the tube radius in the effective viscous drag, is also adequate to fit all previously reported results. The different independent sets of L̇ − f data (9, 20, 21) used by Evans and coworkers are displayed in log scale in Fig. 3. The solid line is the best fit using Eq. 16. Note that η_e is the unique floating parameter and that the result depends very weakly on the precise value for f₀. First, one may check a posteriori that the explored range of pulling velocities indeed matches with the estimated limits V₁ ∼ 0.01μm/s and L̇_c ∼ 100 μm/s (Eq. 18) of the permeation regime. Second, it would be important to discuss the significance of the derived surface viscosity η_e by comparison with previously reported values of effective membrane viscosity η_eff.

Fig. 3.

Collection of velocity–force data for tether extrusion from various cells: neutrophils from refs. 20 (●), 9 (+), and 21 (○); neurons from ref. 10 (▴); OHCs from ref. 23 (▿); and RBCs from refs. 8 (□) and 22 (■). The solid lines are fits using Eq. 16 (or Eq. 17 for OHCs) with νη_eκ² ln(R/r_t) as a unique floating parameter. By taking typical values for ν ∼ 10³ μm⁻², κ ∼ 50 k_BT, and ln(R/r_t) ∼ 5, one finds η_e ∼ 10⁻⁶ Pa·s·m for neutrophils, η_e ∼ 10⁻⁸ Pa·s·m for neurons, η_e ∼ 10⁻⁵ Pa·s·m for OHCs, and η_e ∼ 10⁻⁴ Pa·s·m for RBCs.

Numerous works were devoted to the study of tether growth from various cells, namely red blood cells (RBCs) (5, 6, 8, 22), neuronal growth cones (NGCs) (10, 12), outer hair cells (OHCs) (23). In the vast majority of cases, experiments were aimed at measuring the static force to derive the static adhesion energy, W₀. In some cases, careful measurements of the tether force versus extrusion velocity were carried out. Subsequent analysis was always based on the assumption that the viscous force between membrane and cytoskeleton is linearly related to the velocity (10). Although reported force–velocity traces could generally be well fitted by a linear curve, we tested whether our model, which basically predicts a nonlinear dependence, could be applied. Beside the data relative to neutrophils, we have collected three additional sets of published data corresponding to the three different kinds of cells, namely RBCs (8, 22), OHCs (23), and NGCs (10). In Fig. 3, all L̇ − f data points were replotted in log scale. The solid lines are the best fits using Eq. 16 with η_e as a single floating parameter. A good qualitative agreement is found for each type of cell. However, it might still seem puzzling that these L̇ − f traces can be equally fitted by assuming a linear dependence or a power law. From our viewpoint, the large scatter generally observed in experimental data points is not the main reason. The linear model proposed by Hochmuth et al. (10) is an approximation of our model by assuming that the tube radius remains constant over the range of pulling velocities experimentally explored. Indeed, combination of Eqs. 4, 11, and 12 readily shows that the effective viscosity is given by η_eff = νη_er_t² ln(R/r_t). Although the hypothesis of Hochmuth et al. (10) is not rigorously valid, it may serve as a reasonable approximation over a restricted range of extrusion velocities. Furthermore, by approximately estimating r_t ∼ 10 nm, R ∼ 10 μm, ν ∼ 10³ μm⁻², and κ ∼ 50k_BT, it comes out that η_e ∼ η_eff. Although these parameters are likely to vary from one kind of cell to another, the fitted values of η_e are in relatively good agreement with the η_eff values reported previously (see caption of Fig. 3). As expected, we indeed find that η_e^RBC > η;_e^OHC > η;_e^neutrophil > η;_e^NGC. Thus, the knowledge of the precise values for ν and κ should allow us to convert an effective (apparent) viscosity η_eff into the surface viscosity η_e.

Such an achievement is especially possible in the case of RBCs, which have been extensively characterized from a biochemical and mechanical point of view (24). In this particular case, the bending modulus is known to be κ = 50k_BT, and the bond density is known to be ν ≃ 10³ μm⁻² for the major linkage between the cytoskeleton and the membrane, namely the one mediated by ankyrin that interacts with the cytoskeletal spectrin and the protein band 3 located in the membrane.

The obtained value of η_e ≃ 10⁻⁴ Pa·s·m (100 pN·s/μm) sets a threshold velocity L̇_c ∼ 0.1 μm/s. Beyond the determination of the membrane viscosity, our model also allows us to gain insight in the molecular mechanism of tether extrusion. Indeed, most reported experiments on RBCs were performed at L̇ > L̇_c, which means that the so-called slipping regime was reached. In other words, integral proteins are torn out and dragged into the tube upon extrusion. Although never observed directly in tethers, such a phenomenon has already been evidenced in RBC vesiculation (25) and in membrane spicules of sickled RBCs (26). The description of the slipping contribution to the dynamics of tether growth is beyond the scope of the present paper. Yet, we observe experimentally no significant change of the power law f ∼ L̇^1/3.

The range of ultra-low velocities L̇ < V₁ has never been explored experimentally. Practically, in all reported measurements, the ultra-viscous regime plays a negligible role in the dissipation (see Appendix).

In conclusion, the proposed model aims at reconciliating two different views of tube extrusion from cells, namely a shear thinning behavior of the membrane supported by a phenomenological description and a linear force–velocity relation based on a physical description of the viscous drag between the membrane and the cytoskeleton. By generalizing the physical model in the case where the tube radius depends on the pulling force and extrusion velocity, we were able to describe the shear thinning behavior with simple Newtonian flows.

Moreover, our model provides the basis for a detailed description of membrane–cytoskeleton interactions as a function of the pulling velocity. Different extrusion regimes can be anticipated. At ultra-low velocity, the membrane behaves as an ultra-viscous sheet; at intermediate velocity, it flows through the integral proteins; at ultra-high velocity, the proteins detach from the cytoskeleton. Our predictions for the intermediate permeation regime were critically compared to existing experimental results.

We also believe that our model defines a theoretical framework that could stimulate future experiments to gain insight into the complexity of membrane cytoskeleton interactions.

Appendix

Global Viscous Regime.

If L̇ < V₁, the membrane composed of lipids and cytoskeleton-binding proteins behave like an ultra-viscous sheet of surface viscosity η_p ≅ 2νk_BTτ_off [which is the product of the protein osmotic modulus νk_bT by the characteristic unbinding time τ_off = a/V₁ (28). The radial flow ṙ = L̇r_t/r induces an increase of the membrane tension Δσ_v = 2η_p L̇/r_t (29). The extrusion force is given by Eq. 13 with σ_t = σ + W₀ + Δσ_v, one obtains

In the limit η_p L̇ ≪ f₀, Eq. 19 gives Δf = f − f₀ ≃ 4πη_p L̇, which can be directly derived from the viscous dissipation Δf·L̇ = ∫r_tR 2η_p [(∂ṙ/∂r)² + (ṙ/r)²] 2πrdr = 4πη_p L̇² assuming that the tube size is unperturbed.

In the limit η_p L̇ ≫ f₀, Eq. 19 gives f ≃ 8πη_p L̇. This result can be directly obtained by setting σ_t ≅ Δσ_v in Eq. 13. Now, the tube radius r_t ≃ κ/(4η_p L̇) is strongly perturbed.

Partial Viscous Regime and Permeation.

If L̇ > V₁, the viscous regime occurs over a surface delimited by r_c0 = r_t L̇/V₁ (from the mass conservation) and R, where the local velocity ṙ ≤ V₁. The corresponding increase in membrane tension is Δσ_v = 2η_pV₁/r_c0 = 2η_p V₁²/(r_t L).

Between r_t and r_c0, the lipids flow through the proteins and the corresponding increase in membrane tension Δσ_p is now

The general expression for σ_t is then

This expression shows how the contribution of the viscous regime decreases as soon as L̇ ≥ V₁.

Including Eq. 21 in Eq. 13, we obtain the general formula for the extrusion force:

In most cases (for experimentally accessible extrusion velocities), we have 8πη_p V₁²/L̇ ≪ f₀, which means that the contribution of the ultra-viscous regime is negligible in the energy loss. Eq. 22 thus becomes similar to Eq. 17 if one omits the slight difference in the logarithmic factor.