Mechanical stress induced by the polymerisation of an active gel near a surface

Abstract

Actin flow in the cortical cytoskeleton underneath the cell membrane generates mechanical stresses that shape the cell surface. We study this mechanism using an hydrodynamic model of a compressible active gel polymerising at the membrane and undergoing turnover. We determine how actin flow, density relaxation and friction of actin with the membrane generate stress on a corrugated membrane at the linear order in deformation. Analytical solutions in limiting regimes, combined with finite element methods in the general case, provide a map of normal and tangential stresses as functions of compressibility, interfacial friction and actin turnover, and determine the conditions under which actin polymerisation can render the membrane linearly unstable. The non-linear regime is also briefly discussed.

Figures (8)

• Figure 1: Viscous, compressible actin gel polymerizing on a wavy membrane. Actin polymerizes at a speed at $v_p$ at the membrane and depolymerises at a rate $k_d$ throughout the layer, resulting in an actin density $\rho$ decaying away from the membrane. This sets the layer thickness $h$.
• Figure 2: Log-log plot of the normal stress for $\delta v_{p,q}=0$ as a function of the dimensionless wavenumber $\bar{q}=qh_0$, in the high gel-memrbane friction limit ($\bar{\xi}=10$). In the incompressible limit (Eq. \ref{['stress_incompressible']}, black), the stress exhibits two regimes controlled by the nondimensional parameter $q h_0$: a long wavelength scaling, $\sigma_{nn, q}/2\eta k_d \sim (qh_0)^2$ for $q h_0 \ll 1$ and a short wavelength scaling, $\sigma_{nn, q}/2\eta k_d \sim qh_0$ for $qh_0 \gg 1$. The crossover is highlighted by the black vertical dashed line $\bar{q} =1$. In the fully compressible limit (Eq. \ref{['scaling_compressible_linear']}, grey), an additional friction controlled scale enters, $\bar{\xi} = \xi h_0 / \eta$. As a result the curve follows the same scaling as the incompressible case, but now $q h_0$ crosses another threshold marked at $\bar{q} = 3 \bar{\xi} /4$. Dots are the results of FEM numerics (Appendix \ref{['appendix-fem']}) in both cases.
• Figure 3: Comparison of FEM and analytical predicted stress tensor component $\sigma_{zz}(z)$ (corresponding to $\sigma_{nn}$, Eq.\ref{['normal_stress_friction']}, at the linear level) along the slice (black line) located at a minimum on the membrane deformation ($qx=\pi$). We find near perfect agreement between the simulations and analytics. The parameters are chosen as $\ \bar{\chi} =0,\ \bar{\rho}_0 =1,\ \bar{\xi} =20, \bar{q} =\pi$. Numerical stability is ensured by adding a small diffusive contribution to the transport equations (see Appendix \ref{['appendix-fem']}) with $D=0.01$. All parameters are non-dimensional.
• Figure 4: Stress distribution on the membrane as a function of friction $\xi$ and compressibility $\chi$. The color scale indicates the magnitude of the stress variation over the phase space. The dimensionless compressibility is $\bar{\chi}=\chi/(2\eta k_d)$ and the dimensionless friction is $\bar{\xi} = \xi h_0/\eta$. The two side plots show the stress as a function of the friction for different values of the compressibility, $\bar{\chi}=[0,1,20]$, and the stress as a function of the compressibility for different friction parameters, $\bar{\xi} =[0,5,20]$. Dashed lines are provided purely to guide the eye between data points. Parameters: $\bar{u}_0=0.01, h_0q=\pi.$
• Figure 5: Total stress (gel contribution and membrane restoring forces) for a fully compressible gel ($\chi=0$, \ref{['sigmatot']}) in the absence of actin-membrane friction ($\xi=0$). The stress is shown as a function of $q \lambda$ where $\lambda=\sqrt{\kappa/\gamma}$ is the membrane characteristic length, for increasing values of the curvature-polymerisation coupling strength $\bar{\alpha}=\alpha \eta/(\gamma \lambda) = [0\ \rm{(light\ grey)}, 2\ \rm{(grey)}, 3\ \rm{(black)}]$. The membrane is linearly unstable when the total stress is negative.