A deformation gradient decomposition method for the analysis of the mechanics of morphogenesis

Abstract

A new finite element model is proposed for the analysis of the mechanical aspects of morphogenesis and tested on the biologically well studied gastrulation phenomenon, in particular ventral furrow invagination of the Drosophila melanogaster embryo. A set of mechanisms are introduced in the numerical model, which lead to the observed deformed shapes. We split the total deformation into two parts: an imposed active deformation, and an elastic deformation superimposed onto the latter.The active deformation simulates the effects of apical constriction and apico-basal elongation. These mechanisms are associated with known gene expressions and so in this way we attempt to bridge the well explored signalling pathways, and their associated phenotypes in a mechanical model. While the former have been studied in depth, much less can be said about the forces they produce and the mecha- mechanical hypotheses that generate the necessary folding observed in the invagination process. In particular, we conclude that only certain ratios between both modes (apical constriction and apico-basal elongation) can successfully reproduce the invagination process. The model also supports the idea that this invagination requires the contribution of several mechanisms, and that their redundancy provides the necessary robustness.

Theoretical study of Beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis

“…These include the models for invagination of Davidson et al (1995) and Munoz et al (2007), as well as the finite element model for cardiac looping of Ramasubramanian et al (2006). In addition, Hardin and Cheng (1986) used a model to study secondary invagination during gastrulation in the sea urchin.…”

A 3D finite element model of ventral furrow invagination in the Drosophila melanogaster embryo

“…The cell shaping during this process can thus be regarded simplistically as the contribution of two main, localised deformation modes: (i) apical constriction (wedging), and (ii) apical-basal elongation(columnarisation), resulting in bending of the epithelium (Sweeton et al, 1991;Leptin and Grunewald, 1990;Kam et al, 1991;Muñoz et al, 2007).…”

“…Recently Muñoz et al (2007) developed a 2D finite element method (FEM) model of ventral furrow formation. In this work a deformation gradient decomposition method was used to model the permanent (active) deformations and the (passive) hyperelastic deformations as a local quantity applied to the continuum that schematises the epithelial layer.…”

“…The former is modelled as a rigid sleeve shaped shell constraining the deformation, and the latter imposes a constant volume constraint to the volume within the epithelium. The sliding contact condition is numerically modelled in a different manner to that reported in Muñoz et al (2007). Here, we avoid the numerical instabilities (more critical in 3D analysis) by discretising the vitelline membrane with smooth B-Splines (Muñoz, 2007), and hence no faceted surfaces are obtained.…”

Coupling ALE Technique and Harmonic Parametrization to Describe Concurrent and Successive Elementary Cell Deformations

“…Thus, the passive deformation takes place and the cells react and rearrange themselves by an elastic response of the tissue. In order to consider both deformations, we use the deformation gradient decomposition (Smith, 1993;Rodriguez et al, 1994;Taber, 2007;Barret et al, 2007;Conte et al, 2007;Allena et al, 2010a). Alternative approaches based on active stresses have also been considered (Nobile et al, 2010).…”