Active Nematic Flows over Curved Surfaces

Cell monolayers are a central model system in the study of tissue biophysics. In vivo, epithelial tissues are curved on the scale of microns, and the curvature’s role in the onset of spontaneous tissue flows is still not well understood. Here, we present a hydrodynamic theory for an apical-basal asymmetric active nematic gel on a curved strip. We show that surface curvature qualitatively changes monolayer motion compared with flat space: the resulting flows can be thresholdless, and the transition to motion may change from continuous to discontinuous. Surface curvature, friction, and active tractions are all shown to control the flow pattern selected, from simple shear to vortex chains.

Figure 1

(a) Cells with alignment parameter $h_c>0$ in the free energy (blue cells, e.g., fibroblasts [16]), will align along the direction with the most negative (or least positive) curvature, whereas cells with $h_c<0$ (red cells, e.g., MDCK cells [16]) will align oppositely. (b) The surface geometry: a strip, infinite in $y$, with curvature in the transverse $s$ direction $C_{ss}=C_0\cos 2\pi s/L$, where $L$ is transverse contour length and $C_0>0$, showing single cells aligning according to their alignment parameter. (c)–(f) Typical cell orientation and velocity patterns of (c) passive uniform pattern, (d) passive nonuniform pattern, (e) active single shear flow pattern and (f) active double shear flow pattern. The color code refers to $v_y$, and the red arrows indicate flow directions. Cartesian plots of the orientation and velocity are given in the Supplemental Material, Fig. S1 [31].

Figure 2

Flow patterns and transitions in the zero friction limit ($\xi =0$) obtained from numerical simulations (Supplemental Material [31], Sec. I). (a) $C_0–L$ flow pattern diagram for ${\zeta }_s=4$: no flow (gray, NF); single shear (blue, SS); and double shear (red, DS), and analytical prediction [Eq. (5)] of the continuous NF-SS transition (black solid curve). (b) Constant $C_0=-0.8$ cut through (a) (the dotted box) showing continuous NF-SS transition, and discontinuous SS-DS transition. (c) ${\zeta }_s–C_0$ flow pattern diagram and analytical prediction [Eq. (5)] of the NF-SS transition (solid, continuous; dashed, discontinuous). The two black circular spots separate the continuous transition and the discontinuous transition. The blue dashed curve represents the discontinuous SS-NF (or SS-SS) transition while increasing (or decreasing) ${\zeta }_s$ from a SS pattern (see Supplemental Material [31], Fig. S7). The parameter paths $e1\text{−}e2\text{−}e3$ (e) and $f1\text{−}f2\text{−}f3$ (f) are quenching simulations, as shown in (e),(f). Diagrams in (a),(c) are obtained from an initial NF state. (d) Constant $C_0$ cuts [$C_0=0$ and $C_0=0.4$, dashed boxes in (c)] showing a discontinuous NF-SS transition for the curved case, with thresholdless flows around zero activity. (e) Pattern selection upon varying $C_0$ quasistatically [orange path in (c); see Supplemental Material [31], Sec. I.D] with ${\zeta }_s=-1.5$. See also Supplemental Material [31], Movie S1. (f) Pattern selection upon varying ${\zeta }_s$ quasistatically [magenta path in (c); see Supplemental Material [31], Sec. I.D] with $C_0=-1.6$. See also Supplemental Material [31], Movie S2. The initial SS state ($f1$) was prepared through quenching of $C_0$ [see the point ($f1$) in (c) and (e)]. Parameters other than $\xi$ are those in the Supplemental Material [31], Table 1, including ${\nu }_c=0$, $\zeta =0$, and ${\lambda }_b=0$.

Figure 3

The effect of active bend traction ${\lambda }_b$. Results shown here were obtained from full 2D simulations. (a) Phase diagram of the flow patterns [stable (light gray); single shear (blue); double shear (red) and vortex chain (magenta)] with respect to the active splay traction ${\lambda }_s$ and the active bend traction ${\lambda }_b$. (b) Phase transition regulated by the active splay traction ${\lambda }_s$, which corresponds to the horizontal dash box in (a) with ${\lambda }_b=3.5$. (c),(d) The perturbed double shear flow pattern (c) and the vortex chain pattern (d), which correspond to the dataset shown in (a). In the velocity map, the color code represents $v_y$ and arrows denote velocity vectors; in the orientation map, the color code refers to $\theta$ and lines for orientation directors. Parameters: $\xi =0.6$, $L=8$, and $C_0=0$.