Anisotropy can make a moving active fluid membrane rough or crumpled

TL;DR

This work develops a hydrodynamic theory for anisotropic, inversion-asymmetric moving active permeable membranes described by an anisotropic KPZ equation. Through linear stability analysis, one finds a parameter region of linear stability and a distinct instability leading to stripe-like patterns, while RG analysis reveals a strong-coupling fixed point toward emergent isotropy and 2D KPZ universality, alongside a crumpled regime for strong anisotropy. Direct simulations confirm a robust algebraic rough phase with KPZ scaling exponents and demonstrate a crumpled phase in the unstable region, linking membrane fluctuations to active stress-induced anisotropy. The results provide a framework for understanding how activity and anisotropy govern membrane roughness, with potential implications for actin-driven cellular processes and experimental measurements of membrane tension and fluctuations.

Abstract

We present a hydrodynamic theory of anisotropic and inversion-asymmetric moving active permeable fluid membranes. These are described by an anisotropic Kardar-Parisi-Zhang equation. Depending upon the anisotropy parameters, the membrane is either effectively isotropic and algebraically rough with translational short, but orientational long range order, or unstable, suggestive of membrane crumpling.

Figures (8)

• Figure 1: Snapshots of the height profiles ($L=200$) in the steady states for (a) and (b) algebraic rough phase, with parameters (a) $\nu_x=0.50$, $\nu_y=0.45$, $\overline{\nu}=0.10$, $D=0.005$ and $\lambda=24$, and (b) $\nu_x=0.50$, $\nu_y=0.55$, $\overline{\nu}=0.20$, $D=0.005$ and $\lambda=24$. Snapshots (c) and (d) correspond to the crumpled phase, with parameters (c) $\nu_x=1.0$, $\nu_y=0.20$, $\overline{\nu}=1.0$, $\nu_4=0.10$, $D=1.0$ and $\lambda=2.20$, (with ripples at an arbitrary angle) and (d) $\nu_x=1.0$, $\nu_y=1.0$, $\overline{\nu}=2.0$, $\nu_4=0.10$, $D=1.0$ and $\lambda=2.30$ (with ripples at $\pi/4$). Notice the much smaller fluctuations in the algebraic rough phase, compared to the crumpled phase. (See text).
• Figure 2: (a) Two distinct phases in $\beta$-$\alpha$ plane. White region is the KPZ like putative algebraic rough phase. (b) The RG flow diagram in the $\beta$-$\alpha$-$g$ space. Arrows pointing away from the $\beta$-$\alpha$ plane represent the direction of RG flows, indicating a strong coupling algebraic rough phase. Green regions in (a) and (b) correspond to crumpled phase (see text).
• Figure 3: Log-log plots of $\mathcal{W}$ versus $t$ (a) for $L = 500$ with $\nu_x = 0.50$, $\overline{\nu} = 0.10$, $D = 0.005$, $\lambda = 24$, and different values of $\nu_y$. Inset: linear-scale plot for $L = 300$. (b) For $L = 500$ with $\nu_y = 0.55$ and varying $\overline{\nu}$ (other parameters as in (a)). Inset: plot for $L = 300$ in linear-scale. (c) For $\nu_y = 0.55$, $\overline{\nu} = 0.20$, (other parameters as in (a)) and system sizes $L = 100$ to $500$. The red dashed line indicates a linear fit with slope $\tilde{\beta} = 0.22 \pm 2.6 \times 10^{-5}$. (d) Log-log plot of Saturation width $\mathcal{W}_\text{sat}$ versus $L$ for the same parameters as in (c), with a linear fit yielding $\chi = 0.38 \pm 0.002$ (see text).
• Figure 4: Diagrammatic representations of two point functions.
• Figure 5: Wavevector-shell integration in the one-loop diagrams obtained by perturbative expansions of the action functional (\ref{['action']}), corresponding to the anisotropic KPZ equation. Here, $q_x$ is integrated over the range $-\infty$ to $\infty$, while $q_y$ is integrated over two thin strips $\Lambda/b \leq q_y \leq \Lambda$ and $-\Lambda \leq q_y \leq -\Lambda/b$.