Three-Dimensional Chiral Morphogenesis of Active Fluids

Chirality is an essential nature of biological systems. However, it remains obscure how the handedness at the microscale is translated into chiral morphogenesis at the tissue level. Here, we investigate three-dimensional (3D) tissue morphogenesis using an active fluid theory invoking chirality. We show that the coordination of achiral and chiral stresses, arising from microscopic interactions and energy input of individual cells, can engender the self-organization of 3D papillary and helical structures. The achiral active stress drives the nucleation of asterlike topological defects, which initiate 3D out-of-plane budding, followed by rodlike elongation. The chiral active stress excites vortexlike topological defects, which favor the tip spheroidization and twisting of the elongated rod. These results unravel the chiral morphogenesis observed in our experiments of 3D organoids generated by human embryonic stem cells.

Figure 1

Experimental observation and theoretical model setup. (a) 3D tissue morphogenesis in experiment of hESCs under different induction. Papillary structure observed at day 7 (left), helical structure observed at day 9 (right), and a morphological schematic (middle) are shown. (b) Schematic of the boundary settings in theory. (c),(d) Schematics of active stresses. The ellipses represent cells, and the red arrows represent force dipoles in (c) and moment dipole in (d).

Figure 2

Morphological diagram as the function of achiral and chiral activity (${\zeta }_{\mathrm{a}}$, ${\zeta }_{\mathrm{c}}$). Each snapshot is taken at $t=10000$. Green and orange box denotes the simulated papillary and helical structures, respectively.

Figure 3

Morphological features of two typical structures. (a) The snapshots of papillary structure formation when ${\zeta }_{\mathrm{a}}={\zeta }_{\mathrm{c}}=0.006$. The red curve is the central axis. (b) Snapshots of the helical structure formation when ${\zeta }_{\mathrm{a}}=0.008$ and ${\zeta }_{\mathrm{c}}=0.009$. In (a) and (b), the time points are $t=2000$, 4000, 6000, and 9000, respectively. (c) The central axis during the helical structure formation corresponding to (b). (d),(e) Polarization fields $\mathbf{p}$ corresponding to (a) and (b). The arrows show the direction of $\mathbf{p}$, and the color denotes the magnitude of the $z$ component $p_z$ of $\mathbf{p}$.

Figure 4

Active forces and unit orientation during 3D tissue morphogenesis. (a) Unit orientation and active forces corresponding to the initial stage in state (i), tip spheroidization in state (ii), and spiralization in state (iii). The height of the planes corresponding to the active force fields is 4 (i) and 24 (ii), respectively. (b) The $z$ components of ${\mathbf{F}}_{\mathrm{a}}$ and ${\mathbf{F}}_{\mathrm{c}}$ with respect to the radial coordinate $r$, where ${\zeta }_{\mathrm{a}}={\zeta }_{\mathrm{c}}=0.006$. (c) The radial ($F_r$) and circumferential ($F_{\theta }$) components of ${\mathbf{F}}_{\mathrm{a}}$ and ${\mathbf{F}}_{\mathrm{c}}$ with respect to $r$, where ${\zeta }_{\mathrm{a}}={\zeta }_{\mathrm{c}}=0.006$. (d) The $x$ components of ${\mathbf{F}}_{\mathrm{a}}$ and ${\mathbf{F}}_{\mathrm{c}}$, and the central axis $x_0$ versus $z$, where ${\zeta }_{\mathrm{a}}=0.008$ and ${\zeta }_c=0.009$. The blue dashed line indicates their common extremum. (e) The $y$ components of ${\mathbf{F}}_{\mathrm{a}}$ and ${\mathbf{F}}_{\mathrm{c}}$, and the central axis $y_0$ versus $z$.

Figure 5

(a) The growth rate $\dot{R}$ of the tip spheroid as a function of achiral activity ${\zeta }_{\mathrm{a}}$ and chiral activity ${\zeta }_{\mathrm{c}}$. (b) The dependence of $\dot{R}$ on ${\zeta }_{\mathrm{a}}$ under different ${\zeta }_{\mathrm{c}}$. The dashed line is the fitting by using $({\zeta }_{\mathrm{a}}-{\zeta }_{\mathrm{a}0}{)}^{0.62}$, where ${\zeta }_{\mathrm{a}0}=0.003$. (c) The spiral angle $\alpha$ as a function of ${\zeta }_{\mathrm{a}}$ and ${\zeta }_{\mathrm{c}}$. (d) The dependence of $\alpha$ on ${\zeta }_{\mathrm{c}}$ under different ${\zeta }_{\mathrm{a}}$. The stars at the “shoulder” of the curves indicate the thresholds ${\zeta }_{\mathrm{c}}^{*}$ for spiralization.