Figure 1

Contractile filament bundles are very prominent in cell adhesion. Stress fibers typically connect two focal adhesions and, through their contraction, the cell can probe the mechanical properties of the substrate. The viscoelastic properties of stress fibers can be probed by laser cutting or by cyclic loading through a micromanipulator.Reuse & Permissions

Figure 2

Discretized model. (a) The filament bundle is modeled as a linear chain of Kelvin-Voigt bodies, each characterized by a spring of stiffness $k_{\mathrm{int}}$, a dashpot of viscosity ${\gamma }_{\mathrm{int}}$, and a linear extension $a$. Actomyosin contractility is described by a contractile element with contraction force $F_m$ added to each Kelvin-Voigt body in parallel. Viscoelastic interactions between fiber elements and their surrounding are described by an additional set of external Kelvin-Voigt bodies with stiffness $k_{\mathrm{ext}}$ and viscosity ${\gamma }_{\mathrm{ext}}$. The total fiber length is $L$. $u_n$ denotes the displacement of the $n$th node. (b) Schematic drawing of the solution for the node displacements assuming that both ends are pulled by an external force $F_b>F_s$. Since both terminating nodes are pulled outward, the solution for the displacements is antisymmetric with respect to the center node at $n=0$, which therefore does not move. Thus we obtain the boundary conditions of interest, clamped at $n=0$ and pulled by $F_b$ at $n=N$. (c) The index $n$ starts counting at the center node. The index $j$ starts counting at the node which terminates the fiber at the left.Reuse & Permissions

Figure 3

First oscillation. (a) If $\Gamma <\kappa$, the displacements of inner fiber segments exhibit damped oscillations around their final steady state $u_{\mathrm{ss}}(x)$. Here we show a log-log plot of the time course of the absolute difference $|u(x,t)-u_{\mathrm{ss}}(x)|$ calculated from Eq. (41) for the position $x=26.4$, which is close to the position with maximal amplitude and corresponds to the band $n=7$ in Fig. 6. The inset gives the amplitudes of the first and second oscillation along the fiber, with maxima 236 and 9 nm, respectively. The position $x=26.4$ used here is highlighted as a dashed line. (b) The difference $u(x,t)-u_{\mathrm{ss}}$ at the position $x=26.4$ is shown on a linear scale. Numbering of the extremal values are included for comparison with (a). Parameters for (a) and (b) are as in Fig. 6: $(\kappa ,f_s,\Gamma )=(0.028,0.39,0)$ and $l=42.2$, $a=1\hspace{0.5em}\mu \text{m}$.Reuse & Permissions

Figure 4

Second oscillation. (a) Time course of the absolute difference $|u(x,t)-u_{\mathrm{ss}}|$ on a log-log scale at the position $x=11.0$, where the amplitude of the second oscillation attains its maximum. The inset again shows the maximum amplitude of the first and second oscillation along the fiber, but now the position $x=11.0$ is highlighted as a dashed line. (b) The difference $u(x,t)-u_{\mathrm{ss}}$ at the position $x=11.0$ is shown on a linear scale. Numbering of the extremal values are included for comparison with (a). Parameters used for (a) and (b) are the same as in Figs. 3 and 6.Reuse & Permissions

Figure 5

Comparision of $|u(x,t)-u_{\mathrm{ss}}(x)|$ for the continuum model (solid) and for the discrete model (dashed). The continuum solution is calculated for a fiber of length $l=42$ at position $x=26$. The discrete solution is calculated for a fiber with $N=42$ subunits at node $n=26$. Both solutions were calculated for the same parameters $(\kappa ,\Gamma ,f_s)=(0.028,0,0.39)$ as extracted from experimental data. The two solutions agree well with each other and both models predict oscillations.Reuse & Permissions

Figure 6

Experimental results. (a) GFP-actin stress fiber prior to patterning by photo bleaching, scale bar: $3\hspace{0.5em}\mu \text{m}$. (b) Fiber bleached with stripe pattern. (c)–(e) Stress fiber 1, 30, and 140 s after laser cutting. (f) Time-space kymograph reconstructed from fluorescence intensity profiles along the stress fiber. Band positions are extracted by edge detection (solid lines), scale bars: $30\hspace{0.5em}\text{s}$ and $3\hspace{0.5em}\mu \text{m}$. (g) Model fit to the displacement data of shown bands ($n$ increasing from bottom to top) with initial positions $x_{n=1,2,5,7}=(42.2,39.7,31.8,26.4)\hspace{0.5em}\mu \text{m}$ yields $(\kappa ,f_s,\tau ,\Gamma )=(0.028,0.39,34\hspace{0.5em}\text{s},0.0)$ with $a=1.0\hspace{0.5em}\mu \text{m}$. Note that $\Gamma /\kappa \ll 1$ and that the experimental data provide evidence for the predicted oscillations, because the curves for $n=5$ and 7 show dips after cutting.Reuse & Permissions

Figure 7

Log-log plot of storage modulus (solid) and loss modulus (dashed). Scaling at low and high frequencies is shown for both storage and loss modulus. The parameters used are $(\kappa ,\Gamma )=(0.1,0)$ and $l=30$.Reuse & Permissions