Dynamics of cellular focal adhesions on deformable substrates: consequences for cell force microscopy

Abstract

Cell focal adhesions are micrometer-sized aggregates of proteins that anchor the cell to the extracellular matrix. Within the cell, these adhesions are connected to the contractile, actin cytoskeleton; this allows the adhesions to transmit forces to the surrounding matrix and makes the adhesion assembly sensitive to the rigidity of their environment. In this article, we predict the dynamics of focal adhesions as a function of the rigidity of the substrate. We generalize previous theories and include the fact that the dynamics of proteins that adsorb to adhesions are also driven by their coupling to cell contractility and the deformation of the matrix. We predict that adhesions reach a finite size that is proportional to the elastic compliance of the substrate, on a timescale that also scales with the compliance: focal adhesions quickly reach a relatively small, steady-state size on soft materials. However, their apparent sliding is not sensitive to the rigidity of the substrate. We also suggest some experimental probes of these ideas and discuss the nature of information that can be extracted from cell force microscopy on deformable substrates.

FIGURE 1

The two-layer model: linker proteins in the upper layer connect the acto-myosin stress fibers and the mechanosensitive, lower layer that is anchored to the substrate via integrins.

FIGURE 2

The lower layer is deformed by the tangential component of the stress, that acts along the dash-dot line. The rods have no molecular significance but help to visualize the deformation of the molecular units. The stress-induced tilt is not uniform in the layer, giving rise to a nonzero gradient of tilt.

FIGURE 3

Solution of Eq. 11 for the dynamics of the density profile of the linker proteins for a cell on a rigid substrate. The stress pulls on the FA from left to right. The FA grows and slides for this choice of parameters: μ_bulk = −2.7 k_BT, ΔG = 2.5 k_BT, ε_B = 30 k_BT, λ_xz = 40 k_BT/a³, J = 4.2 k_BT, h = 2a, d = 0.23a, and τ = 2 k_BT.

FIGURE 4

Growth dynamics of FAs assuming a stiff substrate with velocities at the front (v_f) and back (v_b) edges of the cluster of linker proteins, together with the sliding velocity v_sliding = (v_f + v_b)/2 and the growth velocity v_growth = v_f – v_b, as a function of the stress f per unit of thermal stress 1/βa³. On the left-hand side, βτ = 0.5, is chosen so that the velocity at the back edge is in the direction opposite that of the stress. On the right-hand side, βτ = 2 and the back edge always moves in the direction of the stress. The sketches below the graph depict the direction of the velocities at the edges of a focal adhesion as a function of the stress, f. We have chosen μ_bulk = −2.7 k_BT, ΔG = 2.5 k_BT, ε_B = 30 k_BT, λ_xz = 40 k_BT/a³ (this corresponds to 0.7 kPa ≤ λ_xz ≤ 20 kPa for 20 nm ≤ a ≤ 60 nm (25), J = 4.2 k_BT, h = 2a, d = 0.23a).

FIGURE 5

Growth velocity for a stress f = 10 k_BT/a³ as a function of the rigidity of the substrate for three sizes, L, of the adhesion: L = 10a (—), L = 100a (– – –), and L = 1000a (– · –). The other parameters are the same as in Fig. 4. For a substrate with rigidity Λ_xz, the adhesion shrinks (v_growth < 0) when its size exceeds a threshold that is proportional to 1/Λ_xz (see Eq. 18).