Level-sets inspired Lagragian markers.

a) Left Schematics of the level set method. Right Fundamentals of the numerical scheme that shapes S2 into S1. b) Illustration of the method in action. Left Raw data consisting of geometric meshes of single cells spatially organised into the embryo. Center Embryo surface mesh resulting from the application of the level set scheme. Right Rendering of the embryonic surface. c) Tracking of cellular junctions. Left Identification of cellular junctions (red dots). Center Corresponding markers (green dots, defined as vertices on the computed embryonic surface closest to the junctions. Right Relative displacement between junctions and their markers at consecutive timepoints. d) Plot over time of the relative displacement between cellular junctions and their markers.

Strain–rate field describes morphogenesis.

The strain-rate tensor field measures the rate at which morphological changes occur in the embryo as a function of time. The strain-rate tensor field is locally represented as a 3x3 symmetric matrix and is completely determined by its eigenvector fields. a) Heatmap of the eigenvector fields of the strain rate tensor. Each row represents a vector field distinguished by a distinct root color (yellow, pink, white). The gradient from the root color to red represents increasing levels of morphological activity. Top Spatiotemporal dynamics of the first eigenvector field. Middle Spatiotemporal dynamics of the second eigenvector field. Bottom Spatiotemporal dynamics of the third eigenvector field. b) Heatmap of the scalar strain rate field. The gradient from yellow to red depicts regions of increasing morphological activity, while black stands for areas of low morphological activity. The heatmaps show high morphological activity in the invaginating endoderm and zippering neural plate, but also across the embryonic animal during rounds of synchronized division.

Spherical harmonics decomposition of morphogenesis.

a) Example of Spherical harmonics decomposition of the scalar strain rate field mapped to the embryo at t = 4.74 hpf. b) Time evolution of the variance ratios of the main modes of ascidian early morphogenesis (Y00 and Y10). The cell population dynamic is also included in the plot for clarity. c) Time evolution of the coefficients f00 and f10 associated with spherical harmonics (Y00 and Y10). The cell population count is also included in the plot for clarity.

Wavelet analysis highlights multi-timescale modes of morphogenesis.

a) Scalogram resulting from the ricker wavelet transform applied to f00(t) over the whole period covered by the dataset t ∈ [4, 8.6] hpf. b) Scalogram resulting from the ricker wavelet transform applied to f00(t) restricted to the gastrulation period t ∈ [4, 6.3] hpf. The high frequency events highlighted hererepresent time points of synchronized division across the embryo. The dark band in the middle separating two large red regions indicates that there are two phases of invagination characterized by large deformations and a relatively calm transition phase in between. c) Scalogram resulting from the ricker wavelet transform applied to f10(t) restricted to the gastrulation period t ∈ [4, 6.3] hpf. Similar to b), the high frequency events indicate synchronized division in the embryo. d) Scalogram resulting from the ricker wavelet transform applied to f20(t) restricted to endoderm invagination t ∈ [4, 5.1] hpf.

Spectral decomposition of morphogenesis in mutant embryo.

a) Top Ascidian mutant morphogenesis. Bottom Spatiotemporal scalar strain rate field mapped to the mutant surface. b) Time evolution of the coefficients f00 and f10 associated with spherical harmonics (Y00 and Y10). The cell population dynamic is also included in the plot for clarity. c) Wavelet transform applied on f00(t) (top) and f10(t) (bottom).