Model of cellular mechanotransduction via actin stress fibers

Abstract

Mechanical stresses due to blood flow regulate vascular endothelial cell structure and function and play a key role in arterial physiology and pathology. In particular, the development of atherosclerosis has been shown to correlate with regions of disturbed blood flow where endothelial cells are round and have a randomly organized cytoskeleton. Thus, deciphering the relation between the mechanical environment, cell structure, and cell function is a key step toward understanding the early development of atherosclerosis. Recent experiments have demonstrated very rapid ($\sim$100 ms) and long-distance ($\sim$10 $\upmu$m) cellular mechanotransduction in which prestressed actin stress fibers play a critical role. Here, we develop a model of mechanical signal transmission within a cell by describing strains in a network of prestressed viscoelastic stress fibers following the application of a force to the cell surface. We find force transmission dynamics that are consistent with experimental results. We also show that the extent of stress fiber alignment and the direction of the applied force relative to this alignment are key determinants of the efficiency of mechanical signal transmission. These results are consistent with the link observed experimentally between cytoskeletal organization, mechanical stress, and cellular responsiveness to stress. Based on these results, we suggest that mechanical strain of actin stress fibers under force constitutes a key link in the mechanotransduction chain.

Appendix

Results obtained with a single stress fiber (Hwang and Barakat 2012) suggest that stress fiber inertia is negligible, so that wave perturbations in the deformation field are damped by fiber internal viscosity. In support of this notion, the results show that force transmission dynamics are indeed dominated by spatially monotonic deformation of stress fibers. Therefore, the structure of the deformation field does not change significantly in time, and displacement of the fiber can be written as:

$$\begin{aligned}&w_\mathrm{v}(x,t)=a_\mathrm{v}(t)\psi _\mathrm{v}(x), \end{aligned}$$

(20a)

$$\begin{aligned}&w_\mathrm{l}(x,t)=a_\mathrm{l}(t)\psi _\mathrm{l}(x). \end{aligned}$$

(20b)

Substituting Eqs. (20a) and (20b) into Eqs. (1a) and (1b) and integrating these equations over the spatial domain yield:

$$\begin{aligned}&\frac{\sigma _\mathrm{p}A}{L}\underbrace{\left( \int _0^1 \frac{\hbox {d}^2 \psi _\mathrm{v}(\hat{x})}{\hbox {d} \hat{x}^2} \hbox {d}\hat{x}\right) }_{C_{1,v}}a_\mathrm{v}(t) \nonumber \\&\quad +\,\frac{\gamma I}{L^3}\underbrace{\left( -\int _0^1 \frac{\hbox {d}^4 \psi _\mathrm{v}(\hat{x})}{\hbox {d} \hat{x}^4} \hbox {d}\hat{x}\right) }_{C_{2,v}}\frac{\hbox {d}a_\mathrm{v}(t)}{\hbox {d}t}+F_\mathrm{v}=0, \end{aligned}$$

(21a)

$$\begin{aligned}&\frac{EA}{L}\underbrace{\left( \int _0^1 \frac{\hbox {d}^2 \psi _\mathrm{l}(\hat{x})}{\hbox {d} \hat{x}^2} \hbox {d}\hat{x}\right) }_{C_\mathrm{l}}a_\mathrm{l}(t) \nonumber \\&\quad +\,\frac{\gamma A}{L}\underbrace{\left( \int _0^1 \frac{\hbox {d}^2 \psi _\mathrm{l}(\hat{x})}{\hbox {d} \hat{x}^2} \hbox {d}\hat{x}\right) }_{C_\mathrm{l}}\frac{\hbox {d}a_\mathrm{l}(t)}{\hbox {d}t}+F_\mathrm{l}=0, \end{aligned}$$

(21b)

where $\hat{x}$ is defined as x / L.

Equations (20a) and (20b) can be used to relate the displacement of the free end of the fiber to the time functions $a_\mathrm{v}$ and $a_\mathrm{l}$: $w_\mathrm{v}^\mathrm{end}(t)=a_\mathrm{v}(t)\psi _\mathrm{v}(0)$ and $w_\mathrm{l}^\mathrm{end}(t)=a_\mathrm{l}(t)\psi _\mathrm{l}(0)$ and $w_\mathrm{l}(t) =a_\mathrm{l}(t)\psi _\mathrm{l} (0)$. Rearranging equations (21) with $\hat{C}_{1,\mathrm{v}}=C_{1,\mathrm{v}}/\psi _\mathrm{v}(0)$, $\hat{C}_{2,\mathrm{v}}=C_{2,\mathrm{v}}/\psi _\mathrm{v}(0)$ and $\hat{C}_\mathrm{l}=C_\mathrm{l}/\psi _\mathrm{l}(0)$, we obtain the following ordinary differential equations (ODEs) that describe the motion of the free end of the fiber ($x=0$):

$$\begin{aligned}&\frac{\sigma _\mathrm{p}A}{L}\hat{C}_{1,\mathrm{v}}w^\mathrm{end}_\mathrm{v}(t) + \frac{\gamma I}{L^3}\hat{C}_{2,\mathrm{v}}\frac{\hbox {d}w^\mathrm{end}_\mathrm{v}(t)}{\hbox {d}t}+F_\mathrm{v}=0, \end{aligned}$$

(22a)

$$\begin{aligned}&\frac{EA}{L}\hat{C}_\mathrm{l}w^\mathrm{end}_\mathrm{l}(t) + \frac{\gamma A}{L}\hat{C}_\mathrm{l}\frac{\hbox {d}w^\mathrm{end}_\mathrm{l}(t)}{\hbox {d}t}+F_\mathrm{l}=0. \end{aligned}$$

(22b)

An order of magnitude analysis on the three constants $\hat{C}_{1,\mathrm{v}}$, $\hat{C}_{2,\mathrm{v}}$, and $\hat{C}_\mathrm{l}$ reveals that their magnitudes are O(1). We detail the analysis for the case of $\hat{C}_{1,\mathrm{v}}$:

$$\begin{aligned} \hat{C}_{1,\mathrm{v}}=\frac{1}{\psi _\mathrm{v}(0)}\int _0^1 \frac{\hbox {d}^2 \psi _\mathrm{v}(\hat{x})}{\hbox {d} \hat{x}^2} \hbox {d}\hat{x}=\frac{1}{\psi _\mathrm{v}(0)}\frac{\hbox {d} \psi _\mathrm{v}(\hat{x})}{\hbox {d} \hat{x}}\Big |_{\hat{x}=1}, \end{aligned}$$

(23)

given that the boundary condition at $x=0$ imposes that $d \psi _\mathrm{v}(\hat{x})/\hbox {d} \hat{x}|_{\hat{x}=0}=0$. The derivative of $\psi _\mathrm{v}$ at $\hat{x}=1$ can be approximated by $(\psi _\mathrm{v}(1)-\psi _\mathrm{v}(0))/(1-0)$, where $\psi _\mathrm{v}(1)=0$. Substituting this into Eq. 23 yields $\hat{C}_{1,v}= O(1)$.

Because forces associated with prestress, elasticity, and material viscosity act against the direction of the externally applied force, their signs should be negative, and it is reasonable to approximate $\hat{C}_{1,\mathrm{v}}=\hat{C}_{2,\mathrm{v}}=\hat{C}_\mathrm{l}=-1$. Hence, the transverse and longitudinal motions of the free end are governed by the two ODEs given by Eqs. 5a and 5b.