Active Fluid Patterning in Inhomogeneous Environments

Abstract

Active stresses in biological cells and tissues drive many developmental processes. However, increasing experimental evidence suggests that additional mechanical interactions with surrounding material can play a crucial role in guiding these processes. We introduce a minimal model of this scenario and investigate how pattern formation in an active material can be controlled by an inhomogeneous environment. Specifically, we consider an active fluid in which a chemical species regulates local active stresses and is redistributed by the resulting flows. We show that active stress patterns within such a fluid exhibit frictiotaxis and systematically characterize how inhomogeneous external friction affects mechanochemical pattern formation instabilities. We find that hydrodynamic screening plays a crucial role in mediating the cross-talk between friction patterns and active fluid self-organization and identify a mechanochemical frustration mechanism that gives rise to pattern oscillations caused by inhomogeneous friction.

Figures (5)

• Figure 1: Mechanism of friction-guided contractility localization. (a) A region of high stress regulator concentration forms spontaneously and aligns with the friction maximum. ($\mathrm{Pe}{} = 800, \ell_h/L = 1/ 2\pi, \varepsilon=0.9$) (b) Gradients in friction $\gamma(x)$ lead to asymmetric flows into contractile patches (right) compared to homogeneous friction (left). As a result, contractile patches move up friction gradients and eventually localize at friction maxima. Gray dashed lines indicate maximum/minimum flow velocities for the homogeneous friction case. Orange dashed lines show average flow velocity across the domain.
• Figure 2: Modulations of patterning instabilities due to inhomogeneous friction. (a) Dispersion relation with ($r>0$, gray lines) and without ($r=0$, black lines boisPatternFormationActive2011) degradation. Solid lines show the case of homogeneous friction [$\varepsilon=0$, Eq. \ref{['eq:homog-dispersion-relation']}]. Dashed lines show approximate dispersion obtained using perturbation theory with inhomogeneous friction ($\varepsilon\ne0$, friction pattern Eq. (\ref{['eq:fric-function']}) with $n=1$, see SI). Black curves: $r=0, \ell_h/L=1/2 \pi$; gray curves: $r=120,\ell_h/L=1/4\pi$. (b) Critical Péclet number $\mathrm{Pe}{}^*$ as a function of friction inhomogeneity magnitude $\varepsilon$. Solid line shows numerically exact prediction, dashed line shows approximation obtained from perturbation theory, Eq. \ref{['eq:pe-threshold-n1']}. Numerical simulation results of the full model with random initial conditions are shown as blue and red dots. (c) Critical Péclet number for different hydrodynamic length scales $\ell_h/L$. Dashed lines show predictions from perturbation analysis Eq. \ref{['eq:pe-threshold-n1']}. $\mathrm{Pe}$$^*$ becomes independent of $\varepsilon$ when the length scale of hydrodynamic screening $\ell_h$ becomes larger than the friction pattern length scale $\ell_f\sim L$. (d) Critical Péclet number for different friction pattern length scales $\ell_f\sim L/n$ show non-monotonic dependence of $\mathrm{Pe}$$^*$ on $n$. Figures (b)--(d) use $r=0$ (no degradation).
• Figure 3: Unstable concentration modes (blue lines, top) which typically arise in the absence of degradation ($r=0$) lead to flows (orange lines, bottom) whose extrema optimally align with friction minima (vertical dashed lines) when $n=2$ (see Eq. \ref{['eq:fric-function']}). This leads to non-monotonic behavior of critical Péclet number for decreasing friction pattern length scale $\ell_f\sim L/n$ (see Fig. \ref{['fig:dispersion_relation_and_stability']}d).
• Figure 4: Impact of inhomogeneous friction on active fluids with intrinsic pattern length scale selection. (a) Critical Péclet numbers for different friction pattern length scales $\ell_f=L/n$ ($n\in\{1,4,6\}$) when modes with smaller wavelength (here $m=2$, see inset of flow eigenmodes $\delta v$ at $\varepsilon=0$) become unstable first in the homogeneous system. Dashed line indicates regime where the $m=2$ selection is suppressed in favor of a mode with dominant $m=1$ component (see inset $n=1$, $\varepsilon=1$). $\ell_h/L=0.1, r=150$ (b) Total dissipated power $\tilde{\mathcal{D}}$ [see Eq. (\ref{['eq:dissint']})] for parameters from (a) decays most rapidly for friction pattern with $n=1$. (c) Phase diagram comparing dynamics in homogeneous ($\varepsilon=0$, background shading) and inhomogeneous ($\varepsilon=0.9$, colored circles) systems. Dashed line indicates exact stability boundary for $\varepsilon=0$ [$\lambda=0$ in Eq. (\ref{['eq:homog-dispersion-relation']})]. Inhomogeneous friction arrests translational dynamics (blue circles on red shading) and leads to oscillations when steady state and friction patterns become incommensurable (yellow circles on green shading). $n=1$, $r=50$ (d) Kymograph (top) of oscillatory patterning dynamics in which double-peaks (steady-state of the homogeneous system) repeatably form, climb up friction gradients, and collide (bottom).
• Figure S1: Spectrum of $\mathcal{J}$ for two different parameter sets. (a) $n=1$, $\mathrm{Pe}{}=220$, $\alpha = 0.2$, $r=0$. (b) $n=2, \mathrm{Pe}{}=1000, \alpha = 0.2, r=150$. Points show numerically calculated eigenvalues. Full lines show the eigenvalues for $\varepsilon=0$. Dashed lines show lowest-order approximations to the eigenvalue corrections using the perturbation method discussed above. Insets show concentration eigenmodes at the indicated locations.