Organ morphologies are diverse but also conserved under shared developmental constraints among species. Any geometrical similarities in the shape behind diversity and the underlying developmental constraints remain unclear. Plant root tip outlines commonly exhibit a dome shape, which likely performs physiological functions, despite the diversity in size and cellular organization among distinct root classes and/or species. We carried out morphometric analysis of the primary roots of ten angiosperm species and of the lateral roots (LRs) of Arabidopsis, and found that each root outline was isometrically scaled onto a parameter-free catenary curve, a stable structure adopted for arch bridges. Using the physical model for bridges, we analogized that localized and spatially uniform occurrence of oriented cell division and expansion force the LR primordia (LRP) tip to form a catenary curve. These growth rules for the catenary curve were verified by tissue growth simulation of developing LRP development based on time-lapse imaging. Consistently, LRP outlines of mutants compromised in these rules were found to deviate from catenary curves. Our analyses demonstrate that physics-inspired growth rules constrain plant root tips to form isometrically scalable catenary curves.

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Remodeling of multicellular architecture is a critical developmental process for shaping the axis of a bilaterally symmetric animal body and involves coordinated cell–cell interactions and cell rearrangement. In arthropods, the early embryonic process that leads to the segmented body axis varies at the cellular and molecular levels depending on the species. Developmental studies using insect and spider model species have provided specific examples of these diversified mechanisms that regulate axis formation and segmentation in arthropod embryos. However, there are few theoretical models for how diversity in the early embryonic process occurred during evolution, in part because of a limited computational infrastructure. We developed a virtual spherical-shaped multicellular platform to reproduce body axis-forming processes. Each virtual cell behaves according to the cell vertex model, with the computational program organized in a hierarchical order from cells and tissues to whole embryos. Using an initial set of two different mechanical states for cell differentiation and global directional signals that are linked to the planar polarity of each cell, the virtual cell assembly exhibited morphogenetic processes similar to those observed in spider embryos. We found that the development of an elongating body axis is achieved through implementation of an interactive cell polarity parameter associated with edge tension at the cell–cell adhesion interface, with no local control of the cell division rate and direction. We also showed that modifying the settings can cause variation in morphogenetic processes. This platform also can embed a gene network that generates waves of gene expression in a virtual dynamic multicellular field. This study provides a computational platform for testing the development and evolution of animal body patterns.

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