Abstract
During morphogenesis, a featureless convex cerebellum develops folds. As it does so, the cortex thickness is thinnest at the crest (gyri) and thickest at the trough (sulci) of the folds. This observation cannot be simply explained by elastic theories of buckling. A recent minimal model explained this phenomenon by modeling the developing cortex as a growing fluid under the constraints of radially spanning elastic fibers, a plia membrane and a nongrowing sub-cortex (Engstrom et al 2019 Phys. Rev. X 8 041053). In this minimal buckling without bending morphogenesis (BWBM) model, the elastic fibers were assumed to act linearly with strain. Here, we explore how nonlinear elasticity influences shape development within BWBM. The nonlinear elasticity generates a quadratic nonlinearity in the differential equation governing the system’s shape and leads to sharper troughs and wider crests, which is an identifying characteristic of cerebellar folds at later stages in development. As developing organs are typically not in isolation, we also explore the effects of steric confinement, and observe flattening of the crests. Finally, as a paradigmatic example, we propose a hierarchical version of BWBM from which a novel mechanism of branching morphogenesis naturally emerges to qualitatively predict later stages of the morphology of the developing cerebellum.
Highlights
When morphogenesis is viewed through the lens of a physicist, one of the typical mechanistic routes to take is morphoelasticity induced by varying internal stresses [1]
While the linear buckling without bending morphogenesis (BWBM) model addresses the onset of shape change, a follow up question to ask is – what are the limitations of the linear BWBM model in explaining the more dramatic shape changes in the developing cerebellum at later stages of development? As the radial glial cells and the pial membrane become more stretched by the developing crests, the enhanced stretching may lead to detachment of the radial glial cells from the pial membrane or may lead to nonlinear elastic effects
Inspired by cerebellar shape development, we study the effects of nonlinear elasticity, steric confinement, and a branching hierarchy within the BWBM model
Summary
When morphogenesis is viewed through the lens of a physicist, one of the typical mechanistic routes to take is morphoelasticity induced by varying internal stresses [1]. The developing cerebellum is under tension and not under compression as evidenced by both radial and circumferential cuts While these observations cannot be explained by elastic wrinkling theories, all three of these findings can be explained by the linear BWBM model in which a growing cerebellar cortex is fluid-like and the sub-cortex is a nongrowing core [19]. The BWBM model offers an explanation for the length scale invariance of the formation of cerebellum folds to understand the conservation of 8-10 primary lobes across vertebrate species spanning a range of sizes [19]. The linear BWBM model provides a quantitative framework for the onset of shape change that manifest as smooth cortex oscillations in the developing cerebellum.
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