Curvature instability of an active gel growing on a wavy membrane

TL;DR

It is shown that an actin layer polymerising uniformly on the membrane always exerts a stabilising effect that reduces membrane deformation, but curvature-sensitive actin nucleator proteins can render the membrane linearly unstable, giving rise to spontaneous membrane deformation which could initiate extended free-standing cellular protrusion.

Abstract

Cell shape changes are largely controlled by the actin cytoskeleton, a dynamic filament network beneath the plasma membrane. Several cell types can form extended free-standing protrusions not supported by an extracellular substrate or matrix, and regulated by proteins that modulate cytoskeletal dynamics in a way sensitive to the curvature of the cell membrane. We develop a theoretical model for the mechanics of a free-standing viscous actin network growing on a corrugated membrane. The model couples the dynamics of the viscous active gel with membrane deformation and the recruitment of curvature-sensitive actin nucleators. We show that an actin layer polymerising uniformly on the membrane always exerts a stabilising effect that reduces membrane deformation. However, curvature-sensitive actin nucleator proteins can render the membrane linearly unstable, depending on the interplay between membrane and actin dynamics, giving rise to spontaneous membrane deformation which could initiate extended free-standing cellular protrusion.

Figures (4)

• Figure 1: Top: Model for a viscous actin gel polymerizing on a wavy membrane. Balance between polymerization $v_p$ at the membrane and depolymerization $k_d$ yields a layer of finite thickness. Curvature sensitive proteins (purple) locally modulate actin polymerization. Bottom: Two possible scenarios of protein curvature sensing: a curvature-dependent unbinding rate (left) increases the protein bound time in regions of preferred curvature, and/or a spontaneous curvature (right) drives the diffusion of membrane-bound proteins toward regions of preferred curvature.
• Figure 2: Normal stress $\sigma_{nn,q}$ (\ref{['eq:total_stress']}) exerted by a growing actin layer on a sinusoidal membrane as a function of the dimensionless layer thickness $q h_0=qv_p/k_d$, controlled by varying the depolymerization rate $k_d$. For uniform polymerization (black curve) the stress is positive: pulling at the troughs and pushing at the crests of the wavy membrane, stabilizing the flat shape. If the polymerization velocity varies linearly with the local curvature with a coupling strength $\alpha$ (red curve), the stress changes sign and destabilizes modes satisfying $q^2h_0\alpha /v_p>1$. The circles show the numerical results of a finite element method (see SM). Parameters: $qu_q = 2\pi/100$, $\tilde{\alpha} = 10/2\pi$
• Figure 3: Total steady-state membrane stress for fast protein diffusion without recycling. (a) Total normal stress as a function of the wave number $\bar{q}$ for different coupling strengths $\bar{A}$. A negative value indicates a destabilizing effect. (b) Critical coupling strength $\bar{A}^{\ast}$ at which the first unstable mode appears (wavevector $q^*$, red dot in panel (a)) as a function of the layer thickness. Inset shows the value of $q^*$. Parameters: $\bar{H}=1$, $\bar{a}=2$, $\bar{\gamma}=1.5$ and $\bar{h}_0 = 10$ for panel (a).
• Figure 4: Instabilities threshold $A^*$ (the system is unstable if $A>A^*$) as a function of the ratio of protein mobility to recycling rate $\bar{\Lambda}/\bar{k}_{\rm off}^0$. One the left panel, the spontaneous curvature $\bar{H}$ is varied. at fixed curvature-sensitive unbinding rate $\bar{\lambda}_{\rm off}=5$. On the right panel, $\bar{\lambda}_{\rm off}$ is varied at constant $\bar{H} =0.5$. The asymptotic regimes of low and high mobility correspond to Eqs. (\ref{['eq:Acrit_steady_state']},\ref{['eq:Acrit_recycling']}). The instability is eased by protein mobility if $\bar{H}>\bar{a}\rho_0\bar{\lambda}_{\rm off}/\bar{\gamma}$ (see text). Parameters: $\bar{a}=1$, $\bar{b}=0$, $\bar{\gamma}=1$, $\rho_0=0.1$, $\bar{h}_0=10$.