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.

Appendix

Derivation of Eq. (6)

We assume a one-dimensional state of stress in both the SF and FAs in the direction of the SF axis. We further assume, based on the one-dimensional model of FA by Shemesh et al. 24 that at the distal end (i.e., ξ = 0) the FA resist the traction T, and at the proximal end the FA resists the pulling stress σ of the SF. Assuming that the pulling stress from the SF and the traction from the substrate are transmitted to the FA linearly along its length L _FA, it follows that at a distance ξ from the distal end the stress transmitted from the SF to the FA is σ_ξ = (ξ/L _FA)σ and the traction transmitted from the substrate to the FA is T _ξ = (1 − ξ/L _FA)T (see Fig. 1). While both the traction and the SF stress are transmitted to the FA over L _FA, the corresponding contact SF–FA and FA-substrate areas are not the same. This is necessary to provide consistency of the force balance of the free-body diagram shown in Fig. 1.

Mechanical equilibrium demands that at every point ξ of the FA the stresses τ_ξ, σ_ξ, and T _ξ are balanced, i.e., that σ_ξ = τ_ξ + T _ξ, from which we obtain that τ_ξ = σ_ξ − T _ξ = (σ + T)(ξ/L _FA) − T. Using this expression, we obtain the average stress τ within the FA as follows

$$ \tau = {\frac{1}{{L_{\text{FA}} }}}\int\limits_{0}^{{L_{\text{FA}} }} {\tau_{\xi } } d\xi = 0.5(\sigma - T). $$

(A1)

Equilibrium of the whole SF–FA assembly requires that the tensile force due to σ and the force due to the traction are balanced, i.e., that

$$ \sigma A_{\text{SF}} = \int\limits_{{A_{\text{FA}} }} {T_{\xi } dA} = b\int\limits_{0}^{{L_{\text{FA}} }} {T_{\xi } d\xi } , $$

(A2)

where A _SF is the cross-sectional area of the SF, A _FA is the interfacial area of the FA, and b is a constant, representing the width of the FA at the interface with the substrate, i.e., A _FA = bL _FA. By substituting the expression for T _ξ into Eq. (A2), we obtain that

$$ T = 2{\frac{{A_{\text{SF}} }}{{A_{\text{FA}} }}}\sigma . $$

(A3)

By substituting Eq. (A3) into Eq. (A1), we obtain Eq. (6).

Derivation of Eq. (10)

To obtain a relationship between u _n , u ₀, and u _x for simple uniaxial stretching, we follow the steps described in Lazopoulos and Pirentis.19 We assume that the SF is in the substrate xy-plane, at angle θ with respect to the x-axis, and that it carries an initial pre-strain u ₀. The substrate is stretched uniaxially in the direction of the x-axis, with the displacement gradient u _x (Fig. 2). Due to the Poisson’s effect, the substrate also deforms in the y-direction with the displacement gradient u _y  = −νu _x , where ν is the Poisson’s ratio. The deformation gradient tensor F of the SF is a product of the deformation gradient of the substrate F _sub and the deformation gradient F ₀ of the SF due to the pre-strain, where xy-Cartesian components of F _sub and F ₀ are given as follows

$$ {\mathbf{F}}_{\text{sub}} = \left( {\begin{array}{*{20}c} {1 + u_{x} } & 0 \\ 0 & {1 - \nu u_{x} } \\ \end{array} } \right)\quad {\text{and}}\quad {\mathbf{F}}_{0} = \left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {1 + u_{0} } & 0 \\ 0 & 1 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right). $$

(A4)

From Eqs. (A4), we obtain F = F _sub F ₀. Using the right Cauchy-Green strain tensor C = F ^T F, where the superscript T indicates transpose of tensor, we calculate the displacement gradient (u _n ) of the SF as follows $u_{n} = \sqrt {{\mathbf{Cn}} \cdot {\mathbf{n}}} - 1,$ where n = (cos θ, sin θ) is a unit vector parallel with the SF (see Fig. 2), as follows

$$ u_{n} = - 1 + {\frac{{1 + u_{0} }}{\sqrt 2 }}\sqrt {2 + u_{x} [2(1 - \nu ) + u_{x} (1 + \nu^{2} )] + u_{x} [2(1 + \nu ) + u_{x} (1 - \nu^{2} )]\cos 2\theta } . $$

(A5)

For small u _x , i.e., u _x  ≪ 1 and u _x u ₀ ≪ 1, Eq. (A5) can be linearized and we obtain u _n as given by Eq. (10).