Mechanical Stability Determines Stress Fiber and Focal Adhesion Orientation

Abstract

It is well documented in a variety of adherent cell types that in response to anisotropic signals from the microenvironment cells alter their cytoskeletal organization. Previous theoretical studies of these phenomena were focused primarily on the elasticity of cytoskeletal actin stress fibers (SFs) and of the substrate while the contribution of focal adhesions (FAs) through which the cytoskeleton is linked to the external environment has not been considered. Here we propose a mathematical model comprised of a single linearly elastic SF and two identical linearly elastic FAs of a finite size at the endpoints of the SF to investigate cytoskeletal realignment in response to uniaxial stretching of the substrate. The model also includes the contribution of the chemical potential energies of the SF and the FAs to the total potential energy of the SF-FA assembly. Using the global (Maxwell's) stability criterion, we predict stable configurations of the SF-FA assembly in response to substrate stretching. Model predictions obtained for physiologically feasible values of model parameters are consistent with experimental data from the literature. The model shows that elasticity of SFs alone can not predict their realignment during substrate stretching and that geometrical and elastic properties of SFs and FAs need to be included.

Figure 1

A schematic depiction of a stress fiber (SF), focal adhesion (FA), and substrate interaction (top) and the corresponding free-body diagram (bottom). h_FA is the thickness of the FA; σ is the stress within the SF; A_SF is the cross-sectional area of the SF; τ is the average shear stress within the FA; A_FA is the contact area between the FA and the substrate; T is traction at the distal end of the FA. The gray triangles indicate distributions of the SF stress and of the traction at the SF–FA and the substrate–FA interfaces, respectively. While the drawing suggests that the FA is under shear stress, the state of stress in both the FA and the SF is one-dimensional, along the ξ-axis.

Figure 2

A stress fiber (SF)–focal adhesion (FA) assembly lies in the substrate xy-plane. The assembly is oriented at angle θ with respect to the x-axis which is parallel with the direction of substrate stretching. The vector n is the unit vector in the direction of the SF.

Figure 3

The angle of orientation (θ) of the stress fiber-focal adhesion assembly for a wide range of values of parameters A_FA/A_SF and h_FA/L_SF and for E_SF/E_FA = 150, u₀ = 0.1, u_x = 0.1, and ν = 0.35 for case (b) in the text when the total potential includes only the elastic potentials of the SF and FAs and the potential of the contractile stress σ. The gray scale shows the angle θ. Note that the only possible values are 0° or 90°. Calculations were carried out for a discreet number of points which, for better visualization, were sampled logarithmically.

Figure 4

(a) The angle of orientation (θ) of the stress fiber-focal adhesion assembly for a wide range of values of parameters A_FA/A_SF and h_FA/L_SF and for E_SF/E_FA = 150, u₀ = 0.1, u_x = 0.1, and ν = 0.35. The gray scale shows the angle θ for case (c) in the text when the total potential includes the elastic and the chemical potentials of the SF and FAs and the potential of the contractile stress σ. (b) Probability density of θ obtained from the data in (a). Calculations were carried out for a discreet number of points which, for better visualization, were sampled logarithmically (a), whereas for the probability density diagrams the data were sampled linearly (b).

Figure 5

Potential vs. stress fiber orientation (θ) relationships for simple uniaxial stretching during one cycle, for pre-strain u₀ = 0.1, amplitude u_m = 0.05, and substrate Poisson's ratio of ν = 0.35. The potential is scaled with the volume of the stress fiber ${\mathit{\text{V}}}_0^{\text{SF}}$.

Figure 6

Potential vs. stress fiber orientation (θ) relationships for pure uniaxial stretching during one cycle for pre-strain u₀ = 0.1, amplitude u_m = 0.05. The potential is scaled with the volume of the stress fiber ${\mathit{\text{V}}}_0^{\text{SF}}$.

Figure 7

Potential vs. stress fiber orientation (θ) relationships for pure uniaxial stretching during one cycle for pre-strain u₀ = 0.05, amplitude u_m = 0.05. The potential is scaled with the volume of the stress fiber $V_0^{SF}$.