Truss model for stress controlled morphogenesis

Abstract

We resort to the usual decomposition of the deformation gradient into an active and a passive component, and deduce the constitutive law and equilibrium equations when the two components are not independent. In the model described here the active of the deformation is related to the hyperelastic passive part through a control function that simulates a feedback mechanism that has been experimentally observed during embryo development. Using a variational approach, we first write the equations for continua and study the effects of the control function in these equations. We particularise the results for a system of trusses, which allows us to obtain a simplified set of equations. In our derivations, we apply special attention to the conditions that a thermodynamically complaint formulation should satisfy. We particularise these equations and conditions for the relevant elements of the cytoskeleton, namely, microfilaments and microtubules. We apply the model to simulate the shape changes observed during invagination of the Drosophila Melanogaster embryo. As a salient result, the model reveals that the incompressibility constraint of the yolk furnishes a necessary pressure on the epithelium that eventually eases its internalisation.