A mechanical model of actin stress fiber formation and substrate elasticity sensing in adherent cells

Abstract

Tissue cells sense and respond to the stiffness of the surface on which they adhere. Precisely how cells sense surface stiffness remains an open question, though various biochemical pathways are critical for a proper stiffness response. Here, based on a simple mechanochemical model of biological friction, we propose a model for cell mechanosensation as opposed to previous more biochemically based models. Our model of adhesion complexes predicts that these cell-surface interactions provide a viscous drag that increases with the elastic modulus of the surface. The force-velocity relation of myosin II implies that myosin generates greater force when the adhesion complexes slide slowly. Then, using a simple cytoskeleton model, we show that an external force applied to the cytoskeleton causes actin filaments to aggregate and orient parallel to the direction of force application. The greater the external force, the faster this aggregation occurs. As the steady-state probability of forming these bundles reflects a balance between the time scale of bundle formation and destruction (because of actin turnover), more bundles are formed when the cytoskeleton time-scale is small (i.e., on stiff surfaces), in agreement with experiment. As these large bundles of actin, called stress fibers, appear preferentially on stiff surfaces, our mechanical model provides a mechanism for stress fiber formation and stiffness sensing in cells adhered to a compliant surface.

Fig. 1.

A cartoon of stiffness sensing in cells. Left is a schematic diagram of a cell on a soft surface, right is a diagram of a cell on a soft surface (based on figure 1b of ref. 4), where the soft surface has a Young’s modulus of 10 kPa and the stiff surface a modulus of 100 kPa). Adhesion complexes are shown as black dots, stress fibers as thick gray lines. The magnified region, far right, shows how the cell interacts with the surface (the size scale, Black Line, is about 5 μm). Either filipodia, thin projections of actin bundles, or the lamellipodia, a flat, broad expanse of actin, extend via actin polymerization. Connections between the end of these actin filaments appear (adhesion complexes) and the cell pulls on them, through the force-generating properties of nonmuscle myosin II. In some cases, and more frequently on stiff surfaces, the adhesion complexes increase in size and become elongated in the direction of force, and the actin forms a bundle that eventually becomes a stress fiber.

Fig. 2.

A schematic diagram of a cell, showing magnified views of an adhesion complex (AC), nonmuscle myosin II and the cytoskeleton. Nonmuscle myosin II generates force between antiparallel actin filament bundles, one of which is anchored on an AC, the other interacts with the actin cytoskeleton. In both the AC and the cytoskeleton, proteins are drawn as masses on springs in order to indicate how they function in our model.

Fig. 3.

Results of cytoskeleton model simulation. (A) The cytoskeleton model. A constant force is applied in the y direction on a single filament (Black). Three filaments (Dark Gray) are fixed in space with frictionless hinges at one end. Periodic boundary conditions are assumed. (B) Four snapshots from a simulation (full movie is shown in SI Text). As the simulation progresses, a large aggregation of actin filaments appears, oriented in the direction of the applied force (Arrow). (C) Time course of stress fiber formation probability (n_sf) in the absence of actin turnover k_to = 0. Left, demonstrating the time scale of the cytoskeleton. Inset shows stress fiber formation probability for an aggregation of 10 actin filaments as a function of time for two different choices of applied force, filament length, and actin friction parameters. When time is nondimensionalized with the cytoskeleton time scale τ, the two curves fall along a single line. Right, actin aggregations of various sizes show similar formation dynamics. When time is rescaled, the probability of formation of actin aggregations of various size fall along a single line. Thus, we expect that bundles of arbitrary size will be eventually be formed, given enough simulation time. (D) Top, time course of stress fiber formation probability in the presence of actin turnover (k_toτ = 0.0017 not shown). As actin turnover rate increases, the steady-state probability of bundle formation () decreases, indicating that there is a balance between stress fiber formation and breakdown. Bottom, steady-state stress fiber probability as a function of bundle size for four different values of actin turnover.

Fig. 4.

Stress fiber formation as a function of surface stiffness. By choosing parameters in our model, we calculate the steady-state probability of stress fiber formation () of various sizes (N) as a function of the Young’s modulus of the surface. At low stiffness, no stress fibers form, while at high stiffnesses many, large stress fibers form. Inset, simulation results for stress fibers of 10 actin filaments (Gray, ± sd) and an interpolation between the results (Black). Details of the interpolation are in SI Text.