Theoretical Analysis of Membrane Tension in Moving Cells

Abstract

Lateral tension in cell plasma membranes plays an essential role in regulation of a number of membrane-related intracellular processes and cell motion. Understanding the physical factors generating the lateral tension and quantitative determination of the tension distribution along the cell membrane is an emerging topic of cell biophysics. Although experimental data are accumulating on membrane tension values in several cell types, the tension distribution along the membranes of moving cells remains largely unexplored. Here we suggest and analyze a theoretical model predicting the tension distribution along the membrane of a cell crawling on a flat substrate. We consider the tension to be generated by the force of actin network polymerization against the membrane at the cell leading edge. The three major factors determining the tension distribution are the membrane interaction with anchors connecting the actin network to the lipid bilayer, the membrane interaction with cell adhesions, and the force developing at the rear boundary due to the detachment of the remaining cell adhesion from the substrate in the course of cell crawling. Our model recovers the experimentally measured values of the tension in fish keratocytes and their dependence on the number of adhesions. The model predicts, quantitatively, the tension distribution between the leading and rear membrane edges as a function of the area fractions of the anchors and the adhesions.

Introduction

Cell plasma membranes are subject to lateral tension (1). The membrane tension has been suggested to play an important regulatory role in various cellular processes (2) such as endocytosis and exocytosis (3,4), functioning of membrane mechanochemical channels (5), and a mechanical cross-talk between different regions of the cell surface (6,7). The physical factors commonly assumed to produce the membrane tension are the intracellular osmotic pressure and the mechanical forces developing between the membrane and the cytoskeleton (1,2,8–10).

The origin and distribution of tension in membranes of moving cells and the relationship between tension and the intracellular mechanisms driving cell motion has attracted much interest recently (10–12). According to straightforward physical reasoning, the membrane tension distribution in moving cells is expected to be substantially different from that in resting cells. Indeed, because the structural basis of any cell membrane is a lipid bilayer exhibiting properties of a two-dimensional fluid (see Edidin (13) for review), the membrane lateral tension is a two-dimensional analog of pressure existing in ordinary three-dimensional fluids (14). Therefore, under static conditions, the lateral tension must follow the Pascal law, according to which the tension has to be isotropic and homogeneous throughout the whole membrane (6). Yet, membranes of moving cells undergo a complex in-plane flow in addition to the overall rolling and translocation with respect to the external substrates (15). As a result, the tension is expected to behave similarly to pressure in a flowing fluid, which implies existence of tension gradients along the membrane surface.

Recently, experimental efforts of several laboratories have been devoted to measurement of membrane tension in spreading and moving cells by using the method of membrane tether pulling (3,10). In typical experiments performed on fibroblasts (3) or fish keratocytes (10), a bead coated with concanavalin A was bound to the cell membrane and pulled by optical tweezers. The tension was then deduced from measurement of the force applied to the bead by a membrane tether forming between the bead and the cell surface. In the experiments with fish keratocytes, it was verified that the tether pulling did not influence the velocity of the cell movement and that the measured force was independent of the tether length (10). Hence, the tether pulling does not considerably interfere with the cell dynamic behavior.

The membrane tension in the protruding lamellipodium of a spreading fibroblast has been shown to strongly depend on the extent of cell flattening on the substrate, which correlates with exocytosis, myosin contraction, and the capacity of an effective membrane reservoir provided by the plasma membrane of the cell body (3,16). In rapidly moving fish keratocytes, the tension measured, typically, at the cell rear has a characteristic value of a few hundreds of pN/μm and strongly depends on the cytoskeletal forces applied to the membrane and the strength of the cell-substrate adhesion (10). Although more challenging experimentally, the next step would be to measure the tension variations along the membrane surface such as the tension differences between the lamellipodium and the cell center for spreading cells before cell polarization, and between the lamellipodium and the cell rear for polarized crawling cells.

The goal of this work is to analyze and predict theoretically the distribution of membrane tension in a cell moving along a flat substrate. Taking into account that the tension originates from the force applied to the membrane leading edge by the polymerizing actin network (17), we analyze computationally how cell adhesions spanning the membrane and proteins anchoring the lamellipodial actin network to the membrane affect the tension distribution. To assure the quantitative character of the model predictions, we verify a part of the computational results by comparing them with the outcome of measurements on moving keratocytes.

Origin of Forces Determining Membrane Tension

Cell movement is driven by an intracellular network of actin filaments, which undergo directional polymerization against the membrane at the front edge of lamellipodium, a sheetlike membrane protrusion extending from the cell body along the substrate in the direction of motion (18). This network is connected to the extracellular substrate through large multiprotein complexes called cell adhesions (19). The extracellular domains of the cell adhesion proteins are attached to the substrate whereas their cytoplasmic domains dynamically bind actin (20). In addition, the actin network is anchored to the cell membrane itself (Fig. 1 a) by proteins embedded into the membrane matrix and/or by protein-lipid complexes (21,22). The proteinaceous actin-membrane connections referred to below as the membrane anchors are supposed to be freely movable in the plane of a fluid membrane (21). The actin polymerization against the membrane at the lamellipodium front edge generates a force, which drives the membrane forward in the direction of polymerization, and, at the same time, pushes the actin network in the retrograde direction toward the cell body (23). The cell adhesions transmit this force to the external substrate. As a result, the cell moves forward, while the actin network slides backwards undergoing the so-called retrograde flow (24–27).

Figure 1.

Schematic illustration of a moving cell (side views). (a) The lamellipodium is a flat sheetlike membrane protrusion at the cell front, filled by actin filaments that polymerize at its leading edge. The filaments are, generally, attached to the substrate through adhesion complexes, and connected to the membrane via anchor protein complexes. (b) Simplified model of the cell. The polymerization force is counteracted by viscous friction between the anchors and the membrane and by strong friction between the actin network and the adhesions.

There are several forces determining tension values and distribution in the membrane of a moving cell (Fig. 1 b):

• 1.The actin polymerization force pushes the membrane at the front edge.
• 2.The actin flow with respect to the membrane is accompanied by movement of the membrane anchors through the membrane matrix, which generates a viscous membrane flow and the related forces acting on the membrane (15). Similarly, in the course of the cell translocation with respect to the substrate, the ventral membrane undergoes a viscous flow around the cell adhesions, which produces additional forces applied to the membrane.
• 3.It has been suggested that the membrane rear edge is subject to forces resisting cell movement (14) (Fig. 1 b). These rear forces originate from the need to detach from the substrate the residual cell adhesions spanning the rear edge membrane or disconnect the membrane from these adhesions, in case the latter remain bound to the substrate (14). Another source of the rear forces can be the resistance to being crushed of the residual actin network existing at the back of the cell (28).
• 4.The surface of a moving membrane undergoes a friction interaction with the substrate (15) and the surrounding medium (29) (Fig. 1 b).

Because this work aims to understand the tension distribution across the outer membranes of moving cells, we will not relate to the effects of osmotic pressure, which can also exist in stationary cells and produces a (practically) homogeneous contribution to the tension all over the membrane. Thus, we will consider tension produced by the aforementioned forces driving and accompanying the cell motion and analyze its distribution.

Physical Model

We model the cell as a folded membrane spread on a flat substrate and enclosing the actin network (Fig. 1 b). The lower membrane side facing the substrate and the upper one facing the outside solution will be referred to as the ventral and dorsal membrane, respectively, and the membrane fold will be called the leading edge. The leading-edge line is assumed to be straight and much longer than the distance between the ventral and dorsal membranes. As a result, the system is, effectively, one-dimensional and we assume that all its characteristics depend only on the distance from the leading edge measured along the membrane.

The areas of the ventral and dorsal membranes are equal to A, and the membrane has properties of an incompressible two-dimensional fluid characterized by a two-dimensional viscosity η. The membrane anchors are modeled as circular disklike membrane inclusions of radius a, which span the membrane (Fig. 1 b). We assume that the anchors are bound to actin but do not bind the substrate and are movable in the membrane as in a viscous two-dimensional fluid. The cell adhesions connecting the actin network to the substrate are also modeled as circular disklike inclusions of cross-sectional radii a′ that span the ventral membrane (Fig. 1 b). In contrast to the membrane anchors, the extracellular faces of the adhesions are assumed to be firmly attached to the substrate while their intracellular faces undergo an effective frictionlike interaction with the actin network characterized by a friction coefficient β_a-a. In the following, we will often refer to the two types of the membrane inclusions as the anchoring and the adhesion disks.

We use a smeared approximation, in which the discrete localization of the anchors is replaced by a continuous distribution along the membrane characterized by an area fraction σ of the membrane occupied by the anchoring disks. We assume σ to be identical for the ventral and dorsal membrane, and homogeneous along each of them. Similarly, the distribution of the adhesion disks, which are localized only to the ventral membrane, is characterized by a homogeneous area fraction σ′.

The membrane also exhibits interactions with the surrounding medium. The ventral membrane undergoes a viscous friction with the substrate characterized by a friction coefficient β^v_m−s. The dorsal membrane exhibits an interaction with the fluid above the cell. According to (29), this interaction can be described, effectively, as a viscous friction with a friction coefficient β^d_m−s, which is determined by the bulk (three-dimensional) viscosity of the fluid, η_w, and the two-dimensional viscosity of the membrane, η, according to β^d_m−s = η²_w/η (29). For realistic conditions, β^v_m−s is much larger than β^d_m−s, so that β^d_m−s/β^v_m−s ≪ 1. The extracellular faces of the anchoring disks in the ventral and dorsal membranes undergo frictionlike interaction with the substrate and the surrounding fluid characterized, respectively, by the same friction coefficients, β^v_m−s and β^d_m−s, as the membrane.

The actin polymerization force pushing the leading edge forward and the actin network together with the anchoring disks backward will be denoted by $\overrightarrow{F_{\text{pol}}}$, and $-\overrightarrow{F_{\text{pol}}}$, respectively (Fig. 1 b). We assume, for simplicity, that the anchors are distributed symmetrically between the ventral and dorsal membranes and homogeneously along each of the membranes. Moreover, we assume the actin network to be rigid, which means that the velocities of all the anchoring disks attached to the network are equal independently of the anchor positions with respect to each other and to the cell edge. We also assume $\overrightarrow{F_{\text{pol}}}$ to be perpendicular to the leading edge, so that the lateral tension in the ventral and dorsal membranes at the front edge are T_LE = F_pol/2L, where $L=\sqrt{A}$ is the length of the leading edge, $F_{\text{pol}}=\left|\overrightarrow{F_{\text{pol}}}\right|$ is the absolute value of the polymerization force, and the factor ½ takes into account that the force F_pol is equally shared between the ventral and dorsal membranes. The rear force will be denoted by $\overrightarrow{F_r}$. Accordingly, the membrane lateral tension at the rear edge is T_r = F_r/2L, where $F_r=\left|\overrightarrow{F_r}\right|$ is the absolute value of the rear force.

The steady-state dynamics of the system are characterized by three velocities, which can be measured experimentally: the velocity of the cell-edge movement with respect to the substrate, $\overrightarrow{v_m}$; the velocity of the anchoring disks with respect to the substrate $\overrightarrow{v_{\text{ret}}}$; and the velocity of the membrane rolling with respect to the cell edge, $\overrightarrow{v_r}$ (15). The velocity v_m also represents the rate of translocation of the membrane as an entity, and v_ret is also the rate of the actin retrograde flow. The difference of the velocities, $\overrightarrow{v_m}-\overrightarrow{v_{\text{ret}}}$, equals the rate of the actin network elongation due to polymerization. The rolling velocity $\overrightarrow{v_r}$ represents a difference between the translocation velocities of the ventral and dorsal membranes.

The actin polymerization force $\overrightarrow{F_{\text{pol}}}$ depends on the rate of actin polymerization, $\overrightarrow{v_m}-\overrightarrow{v_{\text{ret}}}$, through the force-velocity relationship (30). For our calculations, we use the force-velocity relationship, which has been previously used for rapidly moving keratocytes (7) and described, approximately, by

$$\overrightarrow{v_m}-\overrightarrow{v_{\text{ret}}}=\overrightarrow{v_{\text{max}}}\left[1-{\left(\frac{f_{\text{pol}}}{f_{\text{stall}}}\right)}^8\right],$$ (1)

where $\overrightarrow{v_{\text{max}}}$ is the maximal polymerization velocity reached in the absence of any force exerted on the polymerizing edge of the actin network; f_pol is the absolute value of the force-counteracting polymerization of one actin filament end; and f_stall is the absolute value of the stalling force that, when applied to the polymerizing end of an actin filament, stops its polymerization completely. The parameter f_pol is related to the total polymerization force of the actin network F_pol by f_pol = F_pol/LN_f, where N_f is the number of the actin filament ends polymerizing against a unit length of the leading-edge membrane. For computations, we take v_max = 0.3 μm s⁻¹, f_stall = 6 pN (7,17), and N_f varying between 20 and 200/μm (7).

The rear force $\overrightarrow{F_r}$ must depend, among other factors, also on the velocity of translocation, $\overrightarrow{v_m}$. According to Evans and Ritchie (31) and Bell (32), the dependence of the rupture force for a single adhesion, f_r, on the velocity absolute value, v_m, has to have a logarithmic character,

$$f_r=c_0\ln \left(v_m/\widetilde{v_0}\right),$$ (2)

where $\widetilde{v_0}$ and c₀ are constant parameters depending on the system properties. Because the process of adhesion unbinding is of a stochastic nature, Eq. 2 provides a relationship between the forces required to cause a rate of adhesion unbinding equivalent to the velocity v_m of the rear-edge displacement. The force f_r can be presented as a sum, f_r = f_mem + f_sf, of two contributions: a force, f_mem, which is applied to the adhesion by the stressed membrane and, hence, depends on the membrane tension; and a force, f_sf, determined by factors independent of the membrane tension and velocity. The most essential among the membrane tension-independent factors is the force exerted on the rear adhesions by the acto-myosin stress fibers. This force provides for such cells as fibroblasts a major contribution to the traction stress acting on the substrate underneath the cell rear (33,34), whereas for keratocytes the contributions to the rear traction stress by the stress-fiber and membrane forces may be comparable (27,35).

Based on Eq. 2, the membrane contribution to the adhesion rupture force can be presented as

$$f_{\text{mem}}=c_0\ln \left(v_m/v_0\right),$$ (3)

where the parameter

$$v_0=\widetilde{v_0}\text{exp}\left(\frac{f_{sf}}{c_0}\right)$$

accounts for the effect of the factors independent of the membrane tension, in general, and the stress-fiber force, in particular. Specifically, v₀ describes the velocity of the adhesion disconnection if no membrane force is applied, f_mem = 0, and the disconnection is driven solely by the membrane-independent factors.

We assume that the total force F_r acting on the rear-edge membrane is equal to the average force of a single adhesion f_mem multiplied by the number of adhesions N^r_a at the rear boundary. Because we assume the adhesion area fraction σ′ and the adhesion size to be homogeneous along the ventral membrane, N^r_a is proportional to $\sqrt{{\sigma }^{\text{'}}}$. Taking this into account, the relationship between the rear force F_r and the velocity of the membrane translocation v_m can be presented as

$$F_r=C_0\sqrt{{\sigma }^{\text{'}}}\log \left(v_m/v_0\right),$$ (4)

where the proportionality coefficient relating N^r_a and $\sqrt{{\sigma }^{\text{'}}}$ is included in the constant C₀.

In addition to the adhesion detachment, other factors such as depolymerization and/or breakage of actin filaments at the cell rear (28) might also affect the rear force F_r. However, it has been shown experimentally that the membrane tension at the rear edge, T_r, and, hence, the rear force, F_r, are controlled by the adhesion density (10). Therefore, the filament depolymerization-breakage effect must be either negligible or dependent on the adhesion density. In the latter case, it should be also described by the generic Eq. 4, hence, contributing to the effective characteristic constants C₀ and v₀. Because of a lack of additional knowledge regarding the values of v₀ and C₀, we consider them below as fitting parameters.

Summarizing, the input parameters of the model are f_pol, f_stall, and v_max, determining the polymerization force developed by one actin filament, the number of actin filament ends polymerizing against a unit length of the leading-edge membrane N_f, the area fractions of the anchoring disks, σ, and the adhesions disks, σ′, the friction coefficients of the membrane with the substrate, β^v_m−s, and the fluid above the cell, β^d_m−s, the actin-adhesion friction coefficient, β_a-a, and finally, the parameters v₀ and C₀ determining the rear force. The output of the model will be the determination of the membrane tensions at the leading, T_LE and rear, T_r; the tension distribution along the membrane; and the velocities, v_m, v_r, and v_ret characterizing the system dynamics. As mentioned above, the values of some of the input parameters are taken from the literature or estimated from fitting of the model predictions to the measurement results, while the most crucial of them, σ, σ′, and N_f, will be considered as variables determining the output results.

Results

In the smeared approximation we are using, all the forces acting on the membrane, except for the polymerization force $\overrightarrow{F_{\text{pol}}}$ and the rear force $\overrightarrow{F_r}$, are homogeneously distributed over the areas of the ventral and dorsal membranes. The polymerization force $\overrightarrow{F_{\text{pol}}}$ is concentrated at and evenly distributed along the leading-edge line, whereas the rear force, $\overrightarrow{F_r}$ is concentrated on and evenly distributed along the boundary at the cell rear represented by the rear-edge line. In this case the tension changes linearly along the ventral and dorsal membranes from the value T_LE = F_pol/2L at the leading edge to the value T_r = F_r/2L at the rear edge (where $F_{\text{pol}}=\left|\overrightarrow{F_{\text{pol}}}\right|$, and $F_r=\left|\overrightarrow{F_r}\right|$). Our task is therefore to find F_pol and F_r. Because both F_pol and F_r depend on the velocity of the actin flow with respect to the membrane through the force-velocity relationships in Eqs. 1 and 4 (7,31,32), for any given area fraction of the anchors, σ, and the adhesions, σ′, we have to solve the whole problem for the membrane flow, translocation and rolling.

We first analyze the adhesionless limit, in which the actin network interacts with the membrane through the anchors but does not have a direct interaction with the substrate. This analysis will describe a limiting case of the tension distribution within cells put on low- or nonadhesive substrates (10). Using the obtained results, we then analyze the effects of adhesions on membrane tension. The major technical challenge is related to computation of the membrane flow between the anchoring and adhesion disks for the experimentally relevant values of the disks area fractions σ and σ′. These computations were performed numerically. The full description of the derivations and calculations is given in the Supporting Material.

Membrane tension in the adhesionless limit

For cells with no adhesion, the rear force and, hence, the tension at the rear edge, has to vanish, $\overrightarrow{F_r}=0$; T_r = 0. Hence, our task is to find the tension at the leading-edge T_LE, while, in the smeared approximation, the tension distribution from the leading and the rear edges will follow a linear decay between T_LE to zero. We used specific values of the model parameters based on realistic dimensions and properties of the system components (7). The areas of the ventral and dorsal membranes, which are assumed to have a square shape, are assumed to be A = 100 μm². The radius of an anchoring disk and the membrane viscosity are taken to be a = 5 nm (36,37) and η ≅ 10⁻² pN s μm⁻¹ (38), respectively. The coefficients of the viscous friction between the ventral membrane and the substrate, β^v_m−s, and between the dorsal membrane and the surrounding fluid, β^d_m−s, are estimated to be β^v_m−s ≅ 10⁻² pN s μm⁻³ (39) and β^d_m−s ≅ 10⁻⁴ pN s μm⁻³, respectively. Evaluation of the latter coefficient is based on the relationship β^d_m−s ≅ η_w²/η (29), where η_w ≅ 10⁻³ pN s μm⁻² is the viscosity of water.

The overall velocity of membrane translocation equals zero in the absence of adhesions, v_m = 0, provided that, as assumed in this study, frictions of the anchors and the membrane with the surrounding media are characterized by the same friction coefficients (see (15) and the Supporting Material). The reason for that lies in the interplay between the forces of friction with the external substrate developed, on one hand, by the retrogradely moving anchors and, on the other, by the local membrane flow around the anchors. Because there are no adhesions, these frictions forces are the only factor, which could drive the membrane translocation. Analytical and computational analyses show that movement of every anchor generates membrane backflow within a spatially limited region in the anchor vicinity (15).

The average velocity of this backflow is directed oppositely to that of the anchor motion so that the forces of friction with the substrate generated by the anchor and the locally flowing membrane counteract. According to analytical and numerical computations ((15) and here), these friction forces mutually compensate in the case of equal friction coefficients so that no resultant force appears and, hence, the cell membrane as a whole does not move, v_m = 0. A more detailed discussion of this result is presented in the Supporting Material. Moreover, because the friction force compensation occurs separately for the ventral and dorsal membranes, there is no force driving the membrane rolling and the rolling velocity vanishes, v_r = 0.

Fig. 2 presents the dependence of the tension at the leading edge, T_LE, and the absolute values of the retrograde flow velocity, v_ret, on the area fraction of the anchoring disks, σ, for several densities of the polymerizing filament ends, N_f, at the cell edge. Due to the strongly nonlinear character of the force-velocity relationship (Eq. 1), the velocity of the actin retrograde flow (velocity of the anchors) v_ret, which is equal in this case to the overall velocity of polymerization, is practically constant for low values of σ (Fig. 2 b). Beginning from some characteristic value of σ, the strong viscous interaction between the membrane and the anchoring proteins drives a decay of v_ret (Fig. 2 b).

Figure 2.

Cell movement in the absence of adhesions. The results are presented for several values of the number of polymerizing actin filament ends per micron of the leading edge, N_f. (a) Lateral tension at the membrane leading edge as a function of the membrane area fraction covered by anchoring proteins. The tension at the membrane rear edge equals zero. (b) Absolute value of the velocity of retrograde flow as a function of the membrane area fraction covered by anchoring proteins. Because the overall velocity of the membrane translocation vanishes in this case, the retrograde flow velocity equals the overall velocity of polymerization. To see this figure in color, go online.

The results for the tension at the leading edge, T_LE, are presented in Fig. 2 a for several densities of the polymerizing filament ends at the cell edge N_f. The tension T_LE is determined, through the actin polymerization force, by the viscous interaction between the anchors and the membrane. Therefore, T_LE increases with the anchor area fraction σ and reaches saturation at relatively large σ (Fig. 2 a). The fast increase of the tension T_LE for small σ is due to a strongly nonlinear dependence of the membrane-anchor interaction on the anchor area fraction. The tension saturation at some value of σ (≈8% for cells with N_f = 200/μm) is a consequence of the force-velocity relationship (Eq. 1), according to which the actin polymerization force determining T_LE cannot exceed the maximal value f_stall per filament.

Membrane tension in adhering cells

Three factors determine the difference between the system with adhesions from the nonadhering one considered above:

• 1.The ventral membrane flows around an additional type of disklike inclusion representing the adhesions.
• 2.The adhesions generate a rear force F_r, resisting cell movement with respect to the substrate, which does not exist in nonadhering cells.
• 3.The actin network undergoes a viscous frictionlike interaction with the adhesions, which influences the actin retrograde flow.

To find the parameter values C₀ and v₀ determining the rear force F_r in Eq. 4, we use average values of the tension at the rear edge, T_r, and cell speed, v_m, obtained experimentally for keratocytes with and without treatment with cytochalasin D, which interferes with actin network formation (10). The cytochalasin-treated cells showed reduced speed of v_m = 0.15 ± 0.03 μm/s (mean ± SE; N = 12) compared to the untreated cells characterized by v_m = 0.32 ± 0.03 μm/s (N = 30). The measured rear-edge tension was T_r = 151 ± 12 μm for the cytochalasin-treated compared to T_r = 300 ± 15 μm for the untreated cells. The area fraction of adhesions, assumed to be cytochalasin-independent, was taken to be σ′ ≅ 0.5% as estimated by counting the number of adhesions (Fig. 1 in (40)) multiplied by an average size of 0.2 μm² for an adhesion (E. Barnhart, Stanford, 2013 personal communication). Inserting this data into Eq. 4, along with the connection F_r = 2LT_r, we obtain v₀ = 0.07 μm/s and C₀ = 56 nN. We validate these parameter values by using the experimental results obtained for keratocytes treated with Arp2/3 inhibitor of T_r = 222 ± 21 μm and v_m = 0.22 ± 0.02 μm/s (N = 22) (10). According to our expression, v_m = 0.22 μm/s has to correspond to T_r = 227 pN/μm, which is in a good agreement with the measured value.

The effective actin-adhesion friction coefficient β_a–a, was estimated to be β_a−a = 3.5 · 10⁵ μm⁻³ by comparing the computational results below with the observed retrograde flow in fast-moving keratocytes v_ret ≅ 0.03 μm/s (25,27). The corresponding friction coefficient per adhesion site is ∼0.7 · 10⁵ pN s μm⁻¹.

In the presence of adhesions, the membrane tension at the cell leading edge, T_LE, rear edge, T_r, the velocity of the cell translocation, v_m, the rolling velocity of the membrane, v_r, and the velocity of the actin retrograde flow, v_ret, are determined by the area fractions of both the anchoring disks, σ, and the adhesion disks in the ventral membrane, σ′. We computed separately the dependencies on σ and σ′. Fig. 3 presents the tensions at the leading and rear edges and all the velocities as functions of the area fraction of anchors σ for several densities of the polymerizing actin filament ends at the cell edge, N_f, and for the fixed area fraction of adhesions σ′ ≅ 0.5%. The major effect of adhesions is generation of rear tension, T_r (Fig. 3 a). The difference between the leading- and rear-edge tensions, T_LE – T_r, determines the tension gradient along the membrane of a moving cell. The dependences of T_LE – T_r on σ are presented in the inset of Fig. 3 a.

Figure 3.

Membrane lateral tension and the velocities characterizing cell dynamics in the presence of both anchors and adhesions as functions of the membrane area fraction covered by the anchors. The results are presented for several values of the number of polymerizing actin filament ends per micron of the leading edge, N_f, and for a constant area fraction of the adhesions σ′ = 0.5%. (a) Tension at the cell leading (solid) and rear (dashed) edges. (Inset) Difference between the tensions at the leading and rear edges. This tension difference is compensated by the forces distributed along the membrane, which are dominated by the viscous friction between the membrane and the anchors. (b) Absolute values of the velocities of cell translocation (solid), retrograde flow (dashed), and membrane rolling (dotted). To see this figure in color, go online.

Another adhesion-related effect is the change in the relationships between the velocities characterizing the system. Specifically, the actin polymerization rate is now shared between the retrograde flow rate v_ret and the velocity of the cell translocation, v_m. For σ ≅ 5% corresponding to the experimentally estimated distance between the anchors of ∼40 nm (22), and the estimated above value of σ′ ≅ 0.5%, our model predicts a cell translocation velocity of v_m ≈ 0.22 μm/s, which is in a good agreement with the measurement result of v_m ≈ 0.3 μm/s (10), and a negligibly small rolling velocity, which is also in accord with the observations ((41), K. Keren, Technion, Israel, 2013, unpublished results) (Fig. 3 b). At large area fractions of the anchors, σ ≳ 5%, the viscous interaction between the anchors and the membrane slows down both the translocation and the actin retrograde flow, as was the case in nonadhesive cells.

Reduction of the density of the polymerizing filament ends at the cell leading edge, N_f, is predicted to reduce the tension and the velocities (Fig. 3), which is in a qualitative agreement with the observed effects of cytochalasin-D and Arp2/3 inhibitors in keratocytes (10).

Fig. 4 presents the front and rear tensions, T_LE and T_r, the difference between them, T_LE – T_r, as well as the velocities v_m, v_r, and v_ret as functions of the adhesion area fraction in the ventral membrane σ′ for three fixed values of anchors area fraction σ. As mentioned above, the adhesions are expected to have two opposite effects on the translocation velocity v_m. On one hand, they mediate transmission to the external substrate of the force produced by the actin polymerization and, hence, should accelerate cell translocation. On the other hand, the adhesions at the cell rear have to be detached from the substrate during translocation, and then generate the rear force that slows down cell crawling. The interplay between these two tendencies is expected to result in a nonmonotonic dependence of the cell translocation velocity v_m on the adhesion area fraction σ′.

Figure 4.

The membrane lateral tension and velocities characterizing the cell dynamics in the presence of both anchors and adhesions as functions of the membrane area fraction covered by adhesions. The results are shown for several values of the anchor area fraction. (a) Tension at the cell leading (solid) and rear (dashed) edges. (Inset) Difference between tensions at the leading and rear edges. (b) Absolute values of the velocities of the membrane translocation (solid), retrograde flow (dashed), and membrane rolling (dotted). To see this figure in color, go online.

Indeed, as predicted by our model and presented in Fig. 4 b, the translocation v_m first grows as σ′ increases from zero and then reaches a maximum at a certain value of σ′. Further growth of the adhesion area fraction σ′ results in reduction of the translocation velocity v_m. This behavior is in a qualitative agreement with the results of the experiments where the adhesion amount was changed by variations of RGD ligand on the substrate (10,42).

Discussion

We presented a theoretical model, which enables evaluation of the lateral tension and its gradients in the plasma membrane of a moving cell and understanding of the forces determining the tension values and distribution. The tension arises from the forces generated by actin polymerization against the membrane at the cell leading edge. The major factors setting the tension distribution across the membrane are the interaction between the membrane and the adhesions complexes, which span the ventral membrane and link the actin network to the external substrate, and the interaction between the membrane and proteins anchoring the actin network to the membrane. The interplay between these factors sets the membrane tension at every point along the membrane surface.

Model predictions concerning membrane tension

According to our model, the membrane lateral tension changes along the membrane between the value T_LE at the cell leading- and T_r at the cell rear-edges. The leading-edge tension, T_LE, is set by the force developed by actin network polymerization, F_pol, which, in turn depends through the force-velocity relationship on the speed of the actin flow with respect to the leading-edge membrane. Both the anchors and the adhesions mediate effective friction forces, which resist actin flow, decrease its speed, and hence, increase the polymerization force. Indeed our computations predict a growth of the leading-edge tension, T_LE, with increasing area fractions of the anchors, σ, and of the adhesions, σ′ (Figs. 3 a and 4 a). For the characteristic area fractions of the anchors estimated above, σ ≈ 5%, and adhesions, σ′ ≈ 0.5%, and a realistic density of the actin filament ends pushing the leading-edge membrane, the computed value of T_LE constitutes ∼400 pN/μm.

The tension at the rear edge, T_r, must be smaller than T_LE because a part of the actin polymerization force, F_pol, generating the tension at the leading edge is compensated towards the cell rear by the viscous forces distributed along the membrane and growing with the area fractions of the anchors, σ, and of the adhesions, σ′. Our computations show that for a fixed value of σ′, the compensation effect of the membrane-anchor viscous forces increases faster with σ than does the polymerization force F_pol. As a result, the rear tension T_r is predicted to decrease as a function of the anchor area fraction σ (Fig. 3 a).

The dependence of T_r on the adhesion area fraction σ′ is predicted to be opposite to that on σ, i.e., the rear tension is predicted to increase with growth of σ′ (Fig. 4 a). This means that the polymerization force F_pol rises faster with σ′ than the compensation effect of the membrane-adhesion viscous interaction. The reason for this is that an increase of the adhesion area fraction σ′ results, in addition to the viscous forces, in a growing number of the adhesions, which remain at the rear edge and have to be detached from the substrate in the course of the cell movement. This leads to an additional slowing of cell motion and, consequently, of the actin flow with respect to the leading cell membrane, which reinforces the polymerization force F_pol through the force-velocity relationship.

One of the major predictions of our model concerns the tension difference between the leading and rear edges, T_LE – T_r, setting the tension gradient along the membrane. The tension difference is expected to increase with the area fraction of the anchors, σ (Fig. 3 a, inset). At the same time, T_LE – T_r is predicted to decrease as a function of the area fraction of adhesions σ′ (Fig. 4 a, inset). The latter prediction can be verified experimentally by measuring the leading- and rear-edge membrane tensions in cells moving on substrates with different densities of RGD ligands.

Specifically, for the estimated above values of the anchor σ ∼ 5%, adhesion σ′ ∼ 0.5% area fractions (the latter corresponding to adhesion size of ∼0.2 μm²) (E. Barnhart, Stanford, 2013, personal communication), and number density of 0.025 μm² (40), the rear edge tension, T_r, is predicted to constitute ∼50% of the leading edge tension, T_LE, and the difference between them to be ∼T_LE – T_r ≈ 200 pN/μm.

Model parameters and assumptions

The model is based on a number of parameters, which include the parameters determining the force-velocity relationships for the actin polymerization at the leading edge (Eq. 1) and for the adhesion detachment at the rear edge of the cell (Eqs. 2–4), the friction coefficients among the membrane and the surrounding, β^v_m−s and β^d_m−s, the dimensions of the disklike inclusions representing the anchors of the actin network in the membrane, a, and the cell adhesions, a′.

To evaluate and estimate these parameters we used the results of the membrane tension measurements by the method of membrane tether pulling available for fish keratocytes crawling on flat substrates and of the measurement of the actin retrograde flow (7,10). We performed special computations to test the sensitivity of the model predictions with respect to the parameter values (the results are summarized in Table S1 in the Supporting Material).

The most crucial parameters, which largely determine the membrane tension and the system velocities, are the area fractions of the anchors, σ, and the adhesions, σ′. At the same time, the parameters, such as the sizes of anchors and adhesions, a and a′, and the friction coefficients between the membrane and the surrounding, β^v_m−s and β^d_m−s, have a relatively small effect on the overall tension and velocities. Further, the anchor area fraction σ was taken to be equal for the dorsal and ventral membranes. We tested the possible effects of a difference between the ventral and dorsal values of σ and found it to influence the rolling velocity, but to have only a small effect on the velocity of the overall cell translocation $\overrightarrow{v_m}$ and the tension distribution.

The precision of the parameter determination was limited because the effective tension deduced from the tether pulling experiments includes, in addition to the tension generated by the forces driving the cell motion, contributions from the intracellular osmotic pressure and from the interaction between the membrane bilayer and the cytoskeletal structures underneath the membrane (36). These contributions to the lateral tension were found to constitute no more than ∼25% of the total measured effective tension for fish keratocytes (10). By fitting the parameters values, we neglected this possible 25% correction, which sets the corresponding accuracy of our model predictions. For other cells, the relative contributions of these factors to the effective tension are, probably, different so that the model parameter values need to be readjusted for every cell type.

The model is based on several simplifying assumptions. The strongest assumption is that of the evenly smeared distribution of the anchors along the ventral and dorsal membranes and of the adhesions along the ventral membrane. In reality, both the anchors and adhesions are discrete, and in most cells the adhesions are concentrated within lamella near the cell front. Taking into account the inhomogeneity of the anchor and adhesion distributions would predict, on average, even larger tension gradients along the membrane than those computed here, which is related to a nonlinear dependence of the membrane-anchor and membrane-adhesion viscous interactions on the anchor and adhesions area fractions (15). In addition, we assumed all the anchors to move with the same velocity, implying that the actin network is rigid and continuous throughout the cell. The complex internal dynamics of the real actin network may result in variation of the anchor velocities. The unevenness of the anchor velocities may influence the viscous forces acting between the anchors and the membrane and thus change the local tension gradients but would not alter the overall tension difference between the cell leading and rear edges.

Further, we did our computations assuming the cell front line to be straight, whereas for such cells as keratocytes, it has an arclike shape (7). We ignored the possible effects of variations in cell height assuming the dorsal membrane to be flat and parallel to the substrate. We considered the cell movement to proceed with constant speed whereas for most cells it has a character of periodic protrusion and retractions (43).

Finally, we did not consider possible effects on feedback between membrane tension and membrane transport processes such as endo- and exocytosis (4). All these assumptions may lead to some inaccuracy of the quantitative predictions of the model.