TubULAR: tracking in toto deformations of dynamic tissues via constrained maps

A common motif in biology is the arrangement of cells into tubes, which further transform into complex shapes. Traditionally, analysis of dynamic tissues has relied on inspecting static snapshots, live imaging of cross-sections or tracking isolated cells in three dimensions. However, capturing the interplay between in-plane and out-of-plane behaviors requires following the full surface as it deforms and integrating cell-scale motions into collective, tissue-scale deformations. Here, we present an analysis framework that builds in toto maps of tissue deformations by following tissue parcels in a static material frame of reference. Our approach then relates in-plane and out-of-plane behaviors and decomposes complex deformation maps into elementary contributions. The tube-like surface Lagrangian analysis resource (TubULAR) provides an open-source implementation accessible either as a standalone toolkit or as an extension of the ImSAnE package used in the developmental biology community. We demonstrate our approach by analyzing shape change in the embryonic Drosophila midgut and beating zebrafish heart. The method naturally generalizes to in vitro and synthetic systems and provides ready access to the mechanical mechanisms relating genetic patterning to organ shape change.