Active Nematic Defects and Epithelial Morphogenesis

Inspired by recent experiments that highlight the role of nematic defects in the morphogenesis of epithelial tissues, we develop a minimal framework to study the dynamics of an active curved surface driven by its nematic texture. Allowing the surface to evolve via relaxational dynamics leads to a theory linking nematic defect dynamics, cellular division rates, and Gaussian curvature. Regions of large positive (negative) curvature and positive (negative) growth are colocalized with the presence of positive (negative) defects. In an ex-vivo setting of cultured murine neural progenitor cells, we show that our framework is consistent with the observed cell accumulation at positive defects and depletion at negative defects. In an in-vivo setting, we show that the defect configuration consisting of a bound $+1$ defect state, which is stabilized by activity, surrounded by two $-1/2$ defects can create a stationary ring configuration of tentacles, consistent with observations of a basal marine invertebrate Hydra.

Figure 1

Schematic of our model. Epithelial activity driven by a nematic texture leads to a flow field that drives nematic defects. The defects then induce variations in the intrinsic metric and thence changes in the 3D embedding of the epithelial surface. Image of Hydra in the center, adapted from [28].

Figure 2

Dynamics near defects. Following Eqs. (10) and (11), where $R=-4e^{-\phi }\partial \overline{\partial }\phi$, we show plots of (a) $(d\phi /dt)$ and (b) $(dR/dt)$ for a single $+1/2$ (in red) and a single $-1/2$ defect (in blue), with the activity $\alpha =0$. Top inset: for comparison, we show the experimental growth rate of normalized cell density [17]. Middle and bottom insets: corresponding plots of active contribution for $(d\phi /dt)$ and $(dR/dt)$ for $\varphi =0$, $\alpha =1$. Parameters used are $K=1$, $K^{\prime }=1$, and $\epsilon =1$.

Figure 3

Defect texture and geometry. We numerically integrate Eqs. (8) and (9) (with the substitution ${\partial }_t\to D_t$) to obtain steady state plots of (a) the magnitude of the nematic order parameter $|Q|$ and (b) the curvature density (given by $-4\partial \overline{\partial }\phi$). We note that the sign of the curvature correlates with the sign of the defect, and that the defect configuration is a lattice of $+1$ bound states separated by pairs of $-1/2$ defects. In the inset, we show the profile of the nematic order $|Q|$ (blue) and $\phi$ (red) along the $x$ axis. The profile of $|Q|$, which is dictated by the nematic coherence length, is smaller than the width of the profile of $\phi$ since ${\ell }_{\phi }>\xi$. Parameters for simulations: $\alpha =-0.8$, $K=1$, $K^{\prime }=0$, ${\gamma }_Q={\gamma }_{\phi }=1$, $K_{\phi }=4$, and $\epsilon =2$, in terms of which $\xi =1$, ${\ell }_{R,Q}=1$, and ${\ell }_{\phi }=2$. See text and SM [27], Numerical methods, for details.

Figure 4

Extrinsic geometry. (a) Sketch of the geometry for the tentacle configuration from our simulation. The black dots represent $+1$ defects, the stars represent $-1/2$ defects, and black lines depict the nematic order. Three of the $-1/2$ defects are on the opposite side. (b),(c) Snapshots from simulations of height $u$ of tentacle in real space near a $+1$ defect for early and late times, where insets (adapted from [19]) are snapshots of tentacle formation near a $+1$ defect for early and late times. (d) Plot of the height $h(t)$ at the center of the $+1$ defect as a function of time $t$. Red points are data from simulation and blue curve is the fit $h(t)=h_0[1-\exp (-t/\tau ){]}^{1/2}$, where we find that $h_0=3.87L$ and $\tau =0.01{\tau }_{\phi }$. Initially, $h(t)\propto (L/\xi )(L/{\ell }_{\phi })L\sqrt{(t/{\tau }_{\phi })}$ and $\tau \propto {\tau }_{\phi }$. See SM [27], Algorithm for finding embedding, for details. All plots use rescaled coordinates $x^{\prime }=x/L$, $y^{\prime }=y/L$, and $t^{\prime }=t/{\tau }_{\phi }$.