PMC

Asymptotic analysis of invagination and contraction. (a) Numerical shape resulting from contracting the posterior to a radius r_p < R. (b) Numerical ‘hourglass’ shape resulting from pure invagination. (c) Geometry of contraction with posterior radius r_p < R, resulting in upward motion of the posterior by a distance d. (d) Geometry of pure invagination solution. (e) Asymptotic geometry: in the limit deformations are localized to an asymptotic inner layer of width δ about θ = Θ, where θ = s/R is the angle that the undeformed normal makes with the vertical. In the deformed configuration, this angle has changed to β(θ). (f) Asymptotic invagination: upward motion of the posterior by a distance d requires inward deformations scaling as (δd)^1/2 in the inner layer of width δ. (g) Relation between preferred curvature k and width of invagination λ for a given amount of upward posterior motion d, from asymptotic calculations. (h) Inward rotation Δβ of the midpoint of the invagination with, and without contraction, from asymptotic calculations.

(a) Intraoperative invagination of the pancreatic remnant (P) into the posterior gastric wall (S); (b) postoperative CT scan showing invagination of the pancreatic remnant (P) into the posterior gastric wall (S).

An apico-basal force drives mesoderm invagination. Schemes recapitulate the orientation of the embryos and the laser cuts performed in each experiment. a Longitudinal views of sqhKI[eGFP] control embryos (n = 17), showing the linear shape of the invagination front at early stage. b (top) Longitudinal views of sqhKI[eGFP] embryo ablated in the mid-plane region (red rectangle in the scheme and a dashed magenta line in the figure, n = 12), showing a slight delay in the central region compared to the anterior one, as shown by the outline of the invagination front in magenta at early stage of invagination. (bottom) Scheme of the invagination fronts in the control (white line, as in a) and ablated embryo (magenta line). c Anterior (left) and central (right) transversal views of a mid-plane ablated embryo, at two different times of the invagination. Note that invagination proceeds normally as shown by the V-shaped invagination outlined by a white dotted line (bottom), although a delay in the central region can be observed at early stage (top, compare anterior to central region). d Ventral and transversal views extracted from a movie of a control sqhKI[eGFP] embryos, showing the different step of mesoderm invagination. e The apical surface was cut on the anterior and the posterior sides (blue arrowheads, see left schemes) to isolate the central domain of the mesoderm. Note that the invagination is taking place normally, in term of invagination shape, depth and timing (n = 10). f After isolation of the central domain (apical cuts as described in  e), ablation was performed in the mid-plane of the central region of the mesoderm (red rectangle on the scheme, magenta dashed line in the figure). Note that the invagination is totally prevented when apico-basal forces are impaired in the vast majority of the cases (n = 11/12). e, f Note that apical constriction occurs in both cases, as shown by the curved shape adopted by the ventral surface (white arrowheads). Apico-basal cables are not visible due to low resolution. Scale bar: 10 µm. An anterior, P posterior, V ventral, D dorsal, Ap apical, B basal. See also Supplementary Figs. , and Supplementary Movie 

(a) Timing of the sequence of morphogenetic events at gastrulation. (b) Stretching of posterior endoderm cells by mesoderm invagination in the WT, followed by compression by GBE. Such stretching is shown to be absent in halo sna mutants. Representative of the N=6 embryos observed in each case. (c) Mechanical rescue of posterior endoderm stretching by highly localized magnetic interactions with UML-loaded mesoderm in halo sna mutants. (d) Chemographs of the posterior endoderm of WT, halo sna and magnetically stretched halo sna. Representative of the N=6 embryos observed in each case. (e) Distribution of Myo-II-GFP in sqh-GFP, halo sna sqh-GFP, halo sna sqh-GFP magnetically stretched posterior endoderm, and Fog RNAi-injected halo sna sqh-GFP magnetically stretched posterior endoderm. Representative of the N=6 WT embryos, the N=6 halo sna embryos P=2 × 10⁻³, exact Fisher test), the N=14 halo sna stretched embryos (P<10⁻³) and the N=7 halo sna Fog RNAi-stimulated embryos (P<10⁻³). Scale bars, 5 μm. (f) Invagination (red arrow) in the posterior endoderm of non Rnai-injected (representative of the N=6 embryos observed), sna, bcd, nos, tor Rnai-injected (representative of the N=4 embryos observed, P=2 × 10⁻³), sna, bcd, nos, tor Rnai magnetically stretched posterior endoderm (representative of N=5/8 embryos, P=0.02), and bcd, nos, tor Rnai embryos (N=4, P=0.03). Scale bars, 10 μm.