Stabilization of active tissue deformation by a dynamic signaling gradient

TL;DR

The theoretical results provide quantitative criteria for robust active tissue deformation and explain the lack of gradient-contractile systems in the biological literature, suggesting that the active matter instability acts as an evolutionary selection criterion.

Abstract

A key process during animal morphogenesis is oriented tissue deformation, which is often driven by internally generated active stresses. Yet, such active oriented materials are prone to well-known instabilities, raising the question of how oriented tissue deformation can be robust during morphogenesis. Here we study under which conditions active oriented deformation can be stabilized by the concentration pattern of a signaling molecule, which is secreted by a localized source region, diffuses across the tissue, and degrades. Consistent with earlier results, we find that oriented tissue deformation is always unstable in the gradient-contractile case, i.e. when active stresses act to contract the tissue along the direction of the signaling gradient, and we now show that this is true even in the limit of large diffusion. However, active deformation can be stabilized in the gradient-extensile case, i.e. when active stresses act to extend the tissue along the direction of the signaling gradient. Specifically, we show that gradient-extensile systems can be stable when the tissue is already elongated in the direction of the gradient. We moreover point out the existence of a formerly unknown, additional instability of the tissue shape change. This instability results from the interplay of active tissue shear and signal diffusion, and it indicates that some additional feedback mechanism may be required to control the target tissue shape. Taken together, our theoretical results provide quantitative criteria for robust active tissue deformation, and explain the lack of gradient-contractile systems in the biological literature, suggesting that the active matter instability acts as an evolutionary selection criterion.

Figures (7)

• Figure 1: (A) Hydrodynamic model. A spatially varying source field $s(\bm{r},t)$ secretes a signaling molecule with concentration $c(\bm{r},t)$, which also diffuses and degrades. The gradient of $c$ creates extensile/contractile active anisotropic stresses, which drive viscous flows with velocity $\bm{{\mathrm{v}}}(\bm{r},t)$. These flows affect both source and signaling fields through advection. (B) Typical orders of magnitude for tissue deformation rates $\tilde{v}^0_{xx}$ (top), signal diffusion rates $D/L^2$, where $D$ is the diffusion coefficient and $L$ is the linear dimension of the corresponding tissue (middle), and signal degradation rates $k_\mathrm{d}$ (bottom) from the biological literature. Blue marks are data from the fruit fly Drosophila melanogaster, and orange marks are data from zebrafish, Danio rerio. Tissue deformation rates in the fruit fly were estimated from image time series of: egg chamber Jia2016Alegot2018, dorsal thorax Bosveld2012, pupal wing Etournay2015, hindgut Iwaki2001Johansen2003, renal tubules Saxena2014, and germ-band Bertet2004; and in zebrafish from mesodermal explants Williams2020 and the tailbud Steventon2016. Signaling molecules: Decapentaplegic in the fruit fly Kicheva2007Wartlick2011Romanova-Michaelides2022; Cyclops, Squint, Lefty1 and Lefty2 in zebrafish Muller2013Wang2016.
• Figure 2: Stability of a fixed-size system with a linear source profile: gradient-extensile systems are marginally stable, whereas gradient-contractile systems are unstable. (A) The linear source profile, $s_0(x)=x$ in dimensionless units. In the stationary state, $c_0(x)=s_0(x)$. (B) Maximal perturbation growth rates $\omega_{\mathrm{max}}$ in the $D$--$k_\mathrm{d}$ parameter space show that gradient-extensile systems are marginally stable, i.e. $\omega_\mathrm{max}=0$, highlighted in light gray. (C) $\omega_{\mathrm{max}}$ as a function of signaling molecule diffusion $D$ for a gradient-contractile system with $k_\mathrm{d}=1$, plotted in log-log scaling. (D) $\omega_{\mathrm{max}}$ in the $D$--$k_\mathrm{d}$ parameter space show that gradient-contractile systems are always unstable. The dashed gray line corresponds to the $k_\mathrm{d}=1$ curve plotted in panel C. Panels B-D were created from the result in Appendix \ref{['app:linear source general']}, where the perturbation wave vector $\bm{q}_\mathrm{min} = (0, 2\pi)$ in dimensionless units was used to maximize the perturbation growth rate.
• Figure 3: Stability of a fixed-size system with a localized source profile. (A) The localized source profile $s_0(x)$ from Eq. \ref{['eq:nonlinear source']} for different values of the source width $\mathrm{{w}}$. (B) The localized signaling profile for different values of diffusion $D$. Here, $\mathrm{{w}} = 1/2\pi$ in all cases. (C,D) Snapshots of the source profile from numerical simulations of the perturbed stationary state (with $\mathrm{{w}} = 1/2\pi, D = 10^{-2}, k_{\mathrm{d}}=1$). While the source profile appears to be maintained in the (C) gradient-extensile case, it gets distorted in the (D) gradient-contractile case. (E) Effect of aspect ratio $L_x/L_y$ on maximal perturbation growth rates $\omega_\mathrm{max}$ for different source widths. Gradient-contractile systems (orange) are always unstable, while gradient-extensile systems (green) have in some regions an $\omega_\mathrm{max}$ that can numerically not be distinguished from zero. Parameter values: $k_\mathrm{d} = 1$ and $D = 1$. (F) Effect of the source width $\mathrm{{w}}/L_x$ on maximal perturbation growth rate for different box aspect ratios. Parameter values: $k_\mathrm{d} = 1$ and $D = 1$. (G,H) Color plots of maximal perturbation growth rates for gradient-extensile (G) and gradient-contractile (H) couplings with varying source widths (horizontal axis) and aspect ratios (vertical axis). Marginally stable regions ($\omega_\mathrm{max}=0$) are highlighted in light gray. Solid black curves represent contours, whereas dashed gray lines corresponds to solid curves in panels E,F. Parameter values: $k_\mathrm{d} = 1$ and $D = 1$. (I,J) Maximal perturbation growth rates in the $D$--$k_{\mathrm{d}}$ parameter space for gradient-extensile (I) and gradient-contractile (J) couplings. Numerically marginally stable regions are highlighted in light gray. Parameter values: $L_x/L_y = 1$ and $\mathrm{{w}}/L_x = 0.05$.
• Figure 4: Stability of a freely deforming system with a diffusive signaling molecule, studied in co-deforming coordinates. (A) In lab-frame coordinates, a localized source $s(\bm{r}, t)$ is not stationary when the system deforms affinely, since its width changes proportionally with the system dimension $L_x$ due to advection. (B) However, in co-deforming coordinates, Eqs. \ref{['eq: codef x']} and \ref{['eq: codef y']}, the localized source $s(\bm{\bar{r}}, t)$ remains stationary under affine deformation. (C) Illustration of the affine diffusion-shear instability: diffusion reduces the amplitude $\hat{c}$ of the signaling field, but a larger $L_x$ decreases the effective diffusion coefficient observed in co-deforming coordinates. Thus, a larger $L_x$ leads to a larger $\hat{c}$, which in turn increases active shear and thus $L_x$. (D) Local Lyapunov exponents (LLE) for the gradient-extensile system over time. Blue crosses indicate the numerically computed LLE of the affine diffusion-shear perturbation with the largest growth rate, and the light blue curve displays the analytical prediction according to Eq. \ref{['eq:affine ds instability, growth rate']} in Appendix \ref{['app:localized source, affine diffusion-shear instability, cosine']}. Green squares denote the maximal LLE of all other perturbations. (E) The same results as in panel D, but for the gradient-contractile system. Specifically, black crosses indicate the LLE of the affine diffusion-shear perturbation with the largest growth rate, and orange squares denote the maximal LLE of all other perturbations.
• Figure 5: Instability in a gradient-extensile system with fixed size and a localized source profile. (A) Snapshots of the late-time dynamics of the system in \ref{['fig: fixed nonuniform']}C, showing the source profile at $t = 300$ (left) and $t = 1500$ (right). (B) Perturbation mode with maximum growth rate for system from \ref{['fig: fixed nonuniform']}C and panel A. The color plot shows the signaling molecule perturbation, $\delta c$, and white arrows show the flow field perturbation, $\delta\bm{{\mathrm{v}}}$.