This presentation investigates how organisms reach a specific size, focusing on the role of residual stress, which remains in tissues even after external forces are removed. Currently, the respective role of mechanics and biochemical processes in determining organ size is an open question, and it is unknown how stress is actively built and maintained in tissues during growth. The question here is how growth processes may concurrently determine the stress profile in the bulk of the body as well as the boundary of the body, thus setting its size, independently of perturbations.
The central hypothesis of this talk suggests growth incompatibility as a missing link between tissue growth, size, and residual stress. Growth incompatibility represents the challenge of fitting parts of grown tissue together without voids or overlaps, serving as the geometric "seed" of residual stress. We use the Ricci scalar curvature to approximate incompatibility at both tissue and cell levels.
We propose a theoretical framework where growth incompatibility acts as a geometric regulator for size termination during tissue development. Inspired by vertex models of morphogenesis, we initially explore the hypothesis that the Ricci scalar curvature is prescribed in space and time [1]. Under this assumption, our model successfully reproduces specific experimental observations, such as the characteristic opening patterns observed after tissue cuts in Drosophila wing discs and multicellular spheroids, including the curvature of cut edges and consistent opening patterns following repeated cuts.
Traditional biological growth models rely on the homeostatic (Eshelby-like) stress tensor to define an ideal target state. Any deviation from this state triggers growth and remodeling, aimed at restoring balance between mechanical forces and biological adaptation. However, this homeostatic stress lacks a clear biological interpretation and is often arbitrarily prescribed. To address this limitation, we shift our focus to growth incompatibility, removing the constraint of fixed Ricci curvature used earlier. Instead, we propose a formulation [2] that penalizes deviations from a target incompatibility state, analogous to the Einstein-Hilbert action in General Relativity. This provides a biologically meaningful and physically grounded approach that hints at a link between cellular mechanics and tissue-scale regulation of stress and size.
[1] Erlich, A. and Zurlo, G., 2024. Incompatibility-driven growth and size control during development. Journal of the Mechanics and Physics of Solids, 188, p.105660. DOI: https://doi.org/10.1016/j.jmps.2024.105660
[2] Erlich, A. and Zurlo, G., 2025. The geometric nature of homeostatic stress in biological growth. Journal of the Mechanics and Physics of Solids, p.106155. DOI: https://doi.org/10.1016/j.jmps.2025.106155