A model for boundary-driven tissue morphogenesis

TL;DR

The study addresses how boundary forces from actively deforming neighboring tissues and the embryo's curved geometry can drive morphogenesis of a passive tissue. It develops a minimal elastic-ring model for the hindgut primordium, enforcing inextensibility and area constraints with a fixed AP diameter, and shows that passive boundary effects can produce the characteristic triangular shape via energy minimization $E = (1/2)\oint \kappa(s)^2 ds$. The work demonstrates a two-stage contour kinematics process and a coupled-ring description that reproduce the observed deformations, with curvature gradients on curved embryonic surfaces selecting the observed orientation. This framework provides a general mechanical perspective on how global morphologies arise in development and offers a path to explain blastopore-like shapes across diverse organisms.

Abstract

Tissue deformations during morphogenesis can be active, driven by internal processes, or passive, resulting from stresses applied at their boundaries. Here, we introduce the Drosophila hindgut primordium as a model for studying boundary-driven tissue morphogenesis. We characterize its deformations and show that its complex shape changes can be a passive consequence of the deformations of the active regions of the embryo that surround it. First, we find an intermediate characteristic triangular shape in the 3D deformations of the hindgut. We construct a minimal model of the hindgut primordium as an elastic ring deformed by active midgut invagination and germ band extension on an ellipsoidal surface, which robustly captures the symmetry-breaking into this triangular shape. We then quantify the 3D kinematics of the tissue by a set of contours and discover that the hindgut deforms in two stages: an initial translation on the curved embryo surface followed by a rapid breaking of shape symmetry. We extend our model to show that the contour kinematics in both stages are consistent with our passive picture. Our results suggest that the role of in-plane deformations during hindgut morphogenesis is to translate the tissue to a region with anisotropic embryonic curvature and show that uniform boundary conditions are sufficient to generate the observed nonuniform shape change. Our work thus provides a possible explanation for the various characteristic shapes of blastopore-equivalents in different organisms and a framework for the mechanical emergence of global morphologies in complex developmental systems.

Figures (9)

• Figure 1: The hindgut primordium is bounded by active tissues and rapidly deforms in 15 minutes. (A) brachyury ortholog expression at the gastrula stage across the animal kingdom, in (left to right) the fruit fly Drosophila melanogasterkeenan_dynamics_2022, the sea urchin Lytechnius variegatusgross_role_2001, the lancelet Branchiostoma floridaeyuan_differential_2020, the vertebrate ray-finned zebrafish Danio reriobruce_brachyury_2020, and the amphibian frog Xenopus laevishayata_expression_1999. (B) Dorsal and lateral views of the blastoderm at the onset of gastrulation and 21 minutes later. The cyan signal is a nuclear reporter and the red signal is a nuclear reporter specific to the hindgut (\ref{['mm']}). The germ band, which undergoes in-plane convergent extension, is shaded in white. The ventral furrow and the posterior midgut undergo out-of-plane invagination and are shaded in purple. (C) Dorsal (top) and lateral (bottom) views of the deforming hindgut primordium at five timepoints, showing invagination of the posterior midgut as the hindgut deforms into its characteristic triangular shape. (D) Different views of surface reconstructions of the hindgut primordium from fixed data at timepoints approximated by morphology. Scale bars: $100\,\text{\textmu m}$.
• Figure 2: A minimal physical model reproduces the triangular keyhole shape of the primordial hindgut. (A) The primordial hindgut is modeled as a planar, inextensible elastic ring enclosing an initial area $A = A_0$. Invagination of the midgut reduces the enclosed area to $A$. The observed shape is the shape of lowest energy and symmetric arreaga_elastica_2002veerapaneni_analytical_2009. Additional modes with higher energy also exist and have higher numbers of lobes arreaga_elastica_2002veerapaneni_analytical_2009. (B) The position of the germ band additionally sets the anteroposterior (AP) diameter $d$ of the ring, i.e., the distance between two diametrically opposite points at arclength positions $s=0$, $s=1$. For $d=d_0$, the four shapes of equal lowest energy are AP asymmetric, i.e., asymmetric about the $y$-axis , and include triangular shapes similar to the shape of the primordial hindgut. Additional symmetric and asymmetric shapes are possible as well, but are of higher energies (\ref{['mm']}). (C) Phase diagram of the bifurcation from panel (B) in $(d, A)$ space: The AP asymmetric keyhole shape remains the lowest-energy mode in the shaded region of parameter space as $A$ (midgut invagination) and $d$ (germ band extension) vary. The hatched region is geometrically inaccessible to inextensible deformations. (D) An inextensible elastic ring constrained to lie on a sphere breaks symmetry into one of four shapes with equal energies, analogous to the planar shapes in panel (B), as the area enclosed by the ring is reduced (midgut invagination) while a diameter is fixed (germ band extension). (E) An elastic ring at the posterior pole of an ellipsoid embryo breaks symmetry similarly to the spherical case in panel (D). (F) Symmetry-breaking of an elastic ring at the posterior pole of an ellipsoid after translation to the dorsal side (germ band extension) and reduction of the area enclosed by the ring (midgut invagination): Among the shapes in panels (D), (E), the gradient in curvature consistently selects the triangular shape with the orientation observed in the Drosophila hindgut primordium.
• Figure 3: Data analysis pipeline. (A) The analysis constructs a set of closed space curves ("contours") that are initialized by positions of nuclei within the hindgut primordium and deform with it in time. (B) Light sheet microscopy enables simultaneous imaging of both sides of embryos with fluorescent reporters for nuclei and hindgut. (C) After image fusion and deconvolution (\ref{['mm']}), images are processed using a pixel classifier (ilastik, berg_ilastik_2019) to improve nuclear detection. Scale bars: $200\,\text{\textmu m}$. (D) Nuclei within the hindgut primordium are segmented into spots (top); these spots are tracked semi-automatically using Mastodon tinevez_mastodon_2022 (bottom) to generate a full track for each nucleus in the hindgut primordium. Scale bars: $20\,\text{\textmu m}$. (E) Initial positions of nuclei at the blastoderm stage are mapped into cylindrical axial and angular coordinates $w,\theta$ (inset). (F) Nuclei are binned into contours by their anteroposterior position $w$ in this 2D mapping. (G) Initial nuclear positions from panel (E) colored by the contour to which they are assigned from the binning in panel (F). (H) Contours are fitted using a sequence of splines that update at each timepoint as the nuclei move. Here, the initial contours are overlaid on from panel (G).
• Figure 4: The hindgut primordium deforms in two stages. (A) Shape metrics (contour length, enclosed area, and roundness) plotted against time for a representative embryo, colored by contour (innermost, yellow to outermost, blue). There are two stages: during stage S1 (green), all contours maintain their initial roundness and the lengths and areas of the inner and outer contours decrease and increase, respectively. From $t=15\,\text{min}$ onwards (stage S2, turqoise), the areas enclosed by all contours decrease and the roundness of all contours but the outermost one decreases sharply. Dashed lines, colored by stage, indicate timepoints B1--B5 used in panel (B). Error bars are determined from the standard deviation of a simulated error distribution (\ref{['mm']}). (B) Contour shapes at the timepoints B1--B5 highlighted in panel (A). The violet shading indicates the invaginating posterior midgut (PMG). (C) "Coupled-ring" model of the deformation of circular contours into ellipses (\ref{['mm']}). At time $t$, the middle contour has semi-minor axis ${f_t^x}$ and semi-major axis ${f_t^y}$, and the initial distances ${d_0^\text{i}}, {d_0^\text{o}}$ from the middle to the inner and outer contours have changed to $d_t^\text{i}, d_t^\text{o}$, respectively (inset). (D) Plot of the measured mean distances ${d^\text{i}, d^\text{o}}$ from the middle to the innermost and outermost contours (\ref{['mm']}) against time. Inset: the contours define inner and outer rings used for calculating ${d^\text{i}, d^\text{o}}$. (E) Definition (\ref{['mm']}) of the minor (blue) and major (red) axis lengths ${f_t^x, f_t^y}$, shown for each contour at the initial and final timepoints ${t_0, t_f}$. (F) Plots of the minor and major axis lengths or each contour, normalized by their initial lengths, against time. (G) The "coupled-ring" model (right, \ref{['mm']}) sketched in panel (C) explains the kinematic behaviour of the inner and outer contours during stage S1 (left): If the length (top) or area (bottom) of the middle contour is constant (solid line), the model predicts (dashed lines) that the lengths or areas of the inner and outer contours decrease and increase, respectively, consistently with the data (left).
• Figure S1: Elastic ring in the plane: Bifurcation diagrams. (A) Deformation of an inextensible elastic half-ring of length $\ell/2=1$ in Cartesian axes $(x,y)$. A circular half-ring (gray line) encloses an area $A_0\equiv 1/\pi$ and has diameter $d_0\equiv 2/\pi$. The ring deforms as the area $A$ enclosed by the ring and its diameter $d$ are varied while minimizing its bending energy. The position of a point on the deformed half-ring (green line) is $\boldsymbol{r}(s)=(x(s),y(s))$, where $s\in[0,1]$ is arclength, so that $x(1)-x(0)=d,y(0)=y(1)$. The tangent and normal to the ring are $\boldsymbol{t}(s)$ and $\boldsymbol{n}(s)$, and the tangent angle is $\theta(s)$. Completing the shape of the half-ring into a full ring requires $\theta(0)=\pi/2,\theta(1)=-\pi/2$ (right angles emphasized). Inset equation: definition of the skewness of the deformed shape. (B) Bifurcation diagram of the elastic half-ring for $d=d_0$ in $(p,A)$ space, where $p$ is pressure. Symmetric branches (gray lines) and asymmetric branches (colored lines) bifurcate off the undeformed branch $A=A_0$ at increasing values of $p$. Each asymmetric branch joins a symmetric branch. Circular markers are branch points; the first three branch points BP1, BP2, BP3 are highlighted. Insets: deformed shapes for $A/A_0=0.94$ on different branches; on asymmetric branches, two shapes reflected about $s=1/2$ are shown to emphasize the asymmetry. The first branch to bifurcate (magenta branch) is an asymmetric keyhole shape (similar to the shape of the posterior hindgut). (C) Plot of the first asymmetric branch (shaded surface) in $(d,p,A)$ space. Magenta lines are sections at constant $d$; grey lines are the corresponding symmetric branches linked to these by the branch points BP1 and BP3. These points merge as $d\to d_\text{crit}\approx 1.22$; the first asymmetric branch ceases to exist for $d>d_\text{crit}$. The symmetric branches break up and BP2 disappears for $d\neq d_0$. (D) Plot of the skewness of the first asymmetric branch against $d$ for different $A$.