Role of Turnover in Active Stress Generation in a Filament Network

We study the effect of turnover of cross-linkers, motors, and filaments on the generation of a contractile stress in a network of filaments connected by passive cross-linkers and subjected to the forces exerted by molecular motors. We perform numerical simulations where filaments are treated as rigid rods and molecular motors move fast compared to the time scale of an exchange of cross-linkers. We show that molecular motors create a contractile stress above a critical number of cross-linkers. When passive cross-linkers are allowed to turn over, the stress exerted by the network vanishes due to the formation of clusters. When both filaments and passive cross-linkers turn over, clustering is prevented and the network reaches a dynamic contractile steady state. A maximum stress is reached for an optimum ratio of the filament and cross-linker turnover rates. Taken together, our work reveals conditions for stress generation by molecular motors in a fluid isotropic network of rearranging filaments.

Figure 1

(a) Schematic illustration of a 2D cytoskeletal network (left). Filaments are represented by black arrows, motors by red bars, and cross-linkers by green dots. The network exerts a stress $\sigma$ on the boundaries of the box, arising from tensions acting within the filaments (${\mathbf{f}}_f$) and within the motors (${\mathbf{f}}_m$). Motors move towards the arrowhead of filaments (for actin, the arrowhead corresponds to the barbed end). Network components stochastically exchange with a reservoir (right). (b) Motors move on filaments and rearrange the network on a time scale $\tau$. Cross-linkers and actin filaments turn over on longer time scales ${\tau }_c>\tau$ and ${\tau }_f>\tau$.

Figure 2

Contractile and expansile configurations for one motor and two filaments. (a) Two filaments are attached by fixed cross-linkers to the external network. (b) Possible initial configurations for one motor, two filaments, and one or several cross-linkers, and final configurations. (c) Average strength of the contractile force dipole generated by the molecular motor $⟨d⟩$ in the final configurations (ii),(iii),(v) where the motor does not detach, as a function of the filament length $l_f$. (d) Contractile (force dipole $d>0$) and expansile ($d<0$) configurations.

Figure 3

Stress in a network of motors and filaments, with and without cross-linker turnover. (a) Time evolution of the normalized stress $\sigma /{\sigma }_0$ for ${\tau }_c^{-1}=0$, $N_c=1.2N_f$ (blue), ${\tau }_c=100\tau$, $N_c=1.6N_f$ (red), and ${\tau }_c=100\tau$, $N_c=1.2N_f$ (green) (${\sigma }_0=f_0{l}_m{N}_m/W^2$). (b) Steady-state isotropic stress $\sigma$ (circles) and motor stress ${\sigma }^m$ (diamonds) as a function of the cross-linkers number $N_c$ for ${\tau }_c^{-1}=0$. (c) Fraction of the different configurations in Figs. 2 at steady state as a function of the cross-linker number $N_c$ for ${\tau }_c^{-1}=0$. (d) Isotropic stress $\sigma$ as a function of the cross-linker number $N_c$, for a finite cross-linker lifetime ${\tau }_c=100\tau$, and for several simulation times (${\tau }_{\mathrm{sim}}=300\tau ,600\tau ,1200\tau$, and $2400\tau$). Stress within the motors ${\sigma }_m$ is shown in the inset for ${\tau }_{\mathrm{sim}}=300\tau$ and $2400\tau$. (e) Snapshots of a simulated network with cross-linker turnover ($N_c=1.2N_f$, ${\tau }_c=100\tau$). (f) Stress decay time ${\tau }_{\mathrm{st}}$ as a function of the cross-linker number $N_c$ with cross-linker turnover (${\tau }_c=100\tau$). Solid line, theoretical prediction (see the main text). Other parameters are $N_f=1000$, $N_m=100$, ${\tau }_f^{-1}=0$, $k_{\mathrm{on}}^c{\tau }_c=20$, $l_f=2l_m$, $W=10l_m$, ${\tau }_m=100\tau$, and $k_{\mathrm{on}}^m\tau =20$.

Figure 4

Stress in a network with filament turnover. (a) Time evolution of the stress in a network with cross-linkers and filament turnover for ${\tau }_f=100\tau$, ${\tau }_c=100\tau$ and $N_c/N_f=1.6$. (b) Snapshots of the network evolution corresponding to (a). (c) Steady-state stress $\sigma$ as a function of the number of cross-linkers for ${\tau }_c={\tau }_f=100\tau$ (circles), ${\tau }_c={\tau }_f=10\tau$ (triangles), and ${\tau }_c={\tau }_f=\tau$ (squares). (Inset) Steady-state stress $\sigma$ (circles) and motor stress ${\sigma }^m$ (diamonds) for ${\tau }_c={\tau }_f=100\tau$. (d) Fraction of different configurations (i)–(v) at steady state, as a function of $N_c$ (${\tau }_c=100\tau$, ${\tau }_f=100\tau$). (e) Heat map for contractile stress $\sigma$ as a function of $N_c$ and ${\tau }_c/{\tau }_f$ (${\tau }_c=\tau$). The broken line indicates ${\tau }_f={\tau }_{\mathrm{st}}$. (f) Stress as a function of the ratio of filament and cross-linker turnover rate (${\tau }_c=\tau$, $N_c=1.6N_f$). Other parameters are as in Fig. 3, except that ${\tau }_m^{-1}=0$ and $k_{\mathrm{on}}^f{\tau }_f=20$.