Fig. 3. Decomposition of tissue dynamics into average growth and rescaled dynamics.

A, B Tissue dynamics under a fixed Cartesian coordinate system for real physical space and under the ξ-coordinate system. ξ-coordinates are defined as the spatial coordinates rescaled by average growth, i.e., the spatial average of the volume growth rate or tissue length in a one-dimensional case (A) and that of the deformation gradient tensor in a multi-dimensional case (B). In the ξ -space, tissue dynamics are represented as cell flow reflecting the spatial heterogeneity of local deformation (see the red dotted boxes). C–H Cell trajectories in the ξ -space for Xenopus (C) and chick (D) limb morphogenesis and the positional changes in the proximo-distal (P-D) (E Xenopus; G: chick) or antero-posterior (A-P) (F: Xenopus; H: chick) direction in the ξ -space between an early stage and a stage in which digit patterning is fairly well established (see also Fig. 4C). Arrowheads show the orientations of the flows at t_Xenopus = 52.4 for Xenopus and t_chick = 24 for chick. The colored regions indicate the initially overlapping regions of the limb buds from both species in the ξ-space. For both species, trajectories of the same color indicate trajectories from the same initial position. The cell flows in the ξ coordinate system shown in panels (C) and (D) correspond to the morphogenesis of the region consisting of prospective autopods and zeugopods. Source data are provided as a Source Data file.