Elastic Instabilities Govern the Morphogenesis of the Optic Cup

Because the normal operation of the eye depends on sensitive morphogenetic processes for its eventual shape, developmental flaws can lead to wide-ranging ocular defects. However, the physical processes and mechanisms governing ocular morphogenesis are not well understood. Here, using analytical theory and nonlinear shell finite-element simulations, we show, for optic vesicles experiencing matrix-constrained growth, that elastic instabilities govern the optic cup morphogenesis. By capturing the stress amplification owing to mass increase during growth, we show that the morphogenesis is driven by two elastic instabilities analogous to the snap through in spherical shells, where the second instability is sensitive to the optic cup geometry. In particular, if the optic vesicle is too slender, it will buckle and break axisymmetry, thus, preventing normal development. Our results shed light on the morphogenetic mechanisms governing the formation of a functional biological system and the role of elastic instabilities in the shape selection of soft biological structures.

Figure 1

(a) Simplified schematic of the OC morphogenesis. (b) Thickness change ratio of IOV center wall ${\lambda }_{\mathrm{IOV}}$ as function of the normalized time $t_n$. (c) Invagination depth $D$ normalized by OC horizontal radius $r$ as function of $t_n$. (b) and (c) show comparisons of our model with existing experimental data [9], using $R/h=5$ and $\alpha =40°$ with primary and secondary invagination at $t_n^{1\mathrm{st}}$ and $t_n^{2\mathrm{nd}}$.

Figure 2

(a) Simulation results varying $R/h$ at $\alpha =40°$, considering mass changes. Diamond means the first instability point while triangle and square are the second instability. At $\overline{\theta }=2.51$ (${\overline{\theta }}_b$), the shape-morphing mechanism changes from secondary invagination (triangle) to buckling (square). (b) Representative OC formation process: normal OC (top row) and abnormal OC shape due to buckling (bottom row).

Figure 3

(a) Equivalent natural curvature. (b) Geometrical characteristics on the OOV bending-dominated boundary layer. (c) Characteristic span of each separated cap and OOV.

Figure 4

(a) Phase diagram of instabilities during OC formation for varying $R/h$ at $\alpha =40°$. The blue region denotes the invaginated cup shape, and the lime and red regions are the normal (secondary invagination) and abnormal (buckling) OC, respectively. (b) Phase diagram for varying $R/h$ and $\alpha$. For (a) and (b), the symbols refer to simulation results, with diamonds symbolizing primary invagination, triangles for secondary invagination, and squares for buckling. The lines represent the scaling law of (3), (4), and (5). The black dotted line in (b) shows the buckling transition point ${\overline{\theta }}_b$ of (6).