Extended Data Fig. 10: Crypt bistability mechanism: evaluation of time-dependent effects and extension to in vivo scenario.

A. Theoretical trajectories showing the temporal evolution of crypt morphology, differential tension (that is intrinsic tension in the left panel, and total tension in the right one) and lumen volume, specified as normal morphogenesis (black arrows) and lumen inflation of both bulged and budded organoids (orange arrows). B. Temporal evolution of lumen volume in normal development and suing inflation treatment (left), and corresponding crypt morphologies with various time-scales of mechano-sensing (right). C. Morphological response to lumen inflation for crypts undergoing varying developmental time and showing varying mechano-sensing time-scales. Time-related parameters are normalized by \({\tau }_{{\rm{nd}}}\) (overall time of normal bulged-to-budded evolution). D. Extension of the model to the in vivo setting of an open geometry, where crypt budding can still be modulated and impeded by external force \(t\) generated by other tissues (for example villus). E. Crypt morphology with curvature-sensation (\(n=1\)) in \(\sigma -t\) diagram (left) and corresponding \({\sigma }_{{\rm{c}}}-t\) diagram showing the bistable crypt morphology (right).