The paper describes a mechanical model of epithelial tissue development in Drosophila embryos to investigate a buckling phenomenon called invagination. The finite element method is used to model this ventral furrow formation in 3D by decomposing the total deformation into two parts: an imposed active deformation, and an elastic passive deformation superimposed onto the latter. The model imposes as boundary conditions (i) a constant yolk volume and (ii) a sliding contact condition of the cells against the vitelline membrane, which is interpolated as a B-Spline surface. The active deformation simulates the effects of apical constriction and apico-basal elongation of cells. This set of local cellular mechanisms leads to global shape changes of the embryo which are associated with known gene expressions. Using the model we have tested different plausible hypotheses postulated to account for the mechanical behaviour of epithelial tissues. In particular, we conclude that only certain combinations of local cell shape change can successfully reproduce the invagination process. We have quantitatively compared the model with a 2D model and shown that it exhibits a more robust invagination phenomenon. The 3D model has also revealed that invagination causes a yolk flow from the central region to the anterior and posterior ends of the embryo, causing an accordion-like global compression and expansion wave to move through the embryo. Such a phenomenon cannot be described by 2D models.
"We here assume that this situation is reached when the stresses achieve a target stress, similar to Beloussov's hyper-restoration hypotheses. The particular form of the stress-controlled law has been motivated in our case by the stress profiles obtained in our earlier models where the active kinematic response of the cells was imposed externally (Muñoz et al. 2007; Conte et al. 2008). On the other hand, the nonlinear elastic behaviour of adaptive isotropic chain networks have been studied in Boyce and Arruda (2000), Miehe et al. (2004), and the modelling of oriented chains, commonly found in biology, can be found in Kuhl et al. (2005, 2006). "
[Show abstract][Hide abstract] ABSTRACT: A set of equilibrium equations is derived for the stress-controlled shape change of cells due to the remodelling and growth of their internal architecture. The approach involves the decomposition of the deformation gradient into an active and a passive component; the former is allowed to include a growth process, while the latter is assumed to be hyperelastic and mass-preserving. The two components are coupled with a control function that provides the required feedback mechanism. The balance equations for general continua are derived and, using a variational approach, we deduce the equilibrium equations and study the effects of the control function on these equations. The results are applied to a truss system whose function is to simulate the cytoskeletal network constituted by myosin microfilaments and microtubules, which are found experimentally to control shape change in cells. Special attention is paid to the conditions that a thermodynamically consistent formulation should satisfy. The model is used to simulate the multicellular shape changes observed during ventral furrow invagination of the Drosophila melanogaster embryo. The results confirm that ventral furrow invagination can be achieved through stress control alone, without the need for other regulatory or signalling mechanisms. The model also reveals that the yolk plays a distinct role in the process, which is different to its role during invagination with externally imposed strains. In stress control, the incompressibility constraint of the yolk leads, via feedback, to the generation of a pressure in the ventral zone of the epithelium that eventually eases its rise and internalisation.
Biomechanics and Modeling in Mechanobiology 08/2010; 9(4):451-67. DOI:10.1007/s10237-009-0187-9 · 3.15 Impact Factor
"We here assume that this situation is reached when the stresses achieve a target stress, similar to Beloussov's hyper-restoration hypotheses . The particular form of the stress controlled law has been motivated in our case by the stress profiles obtained in our earlier models where the active kinematic response of the cells was imposed externally (Muñoz et al., 2007; Conte et al., 2008). On the other hand, the non-linear elastic behaviour of adaptive isotropic chain networks have been studied in Boyce and Arruda (2000); Miehe et al. (2004), and the modelling of oriented chains, commonly found in biology , can be found in Kuhl et al. (2005, 2006). "
[Show abstract][Hide abstract] ABSTRACT: The mechanical analysis of soft tissues in biomechanics has experienced increasing progress during the last decade. Part of this success is due to the development and application of some techniques of continuum mechanics, in particular, the decomposition of the deformation gradient, and the introduction of mass, density or volume changes in the reference configuration. Resorting to the common terminology employed in the literature, the changes in biomechanical processes may be classified as growth (change of mass), remodelling (change of density or other material properties such as fibre orientation) or morphogenesis (change of shape). Although the use of those concepts in bone and cardiovascular analysis is well extended, their use in morphogenesis during embryo development has been far less studied. The reasons for this fact may be found in the large shape changes encountered during this process, or the complexity of the material changes involved. In this chapter we develop a general framework for the modelling of morphogenesis by introducing a growth process in the structural elements of the cell, which in turn depends on the stress state of the tissue. Some experimental observations suggest this feedback mechanism during embryo development, and only very recently this behaviour has started to be simulated. We here derive the necessary equilibrium equations of a stress controlled growth mechanism in the context of continuum mechanics. In these derivations we assume a free energy source which is responsible for the active forces during the elongation process, and a passive hyperelastic response of the material. In addition, we write the necessary conditions that the active elongation law must satisfy in order to be thermodynamically consistent. 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 embryo morphogenesis in truss element. As a salient result, the model reveals that by imposing boundary stress conditions, unbounded elongation would be obtained. Therefore, either prescribed displacements or cross-links between fibres are necessary to reach a homeostatic state.
"There are some aspects of invagination which can only be modelled by using a fully 3D model, as shown in Conte et al (2008). However, for the purposes of this analysis, we have constructed a circular 2D model that closely approximates 2D- cross sections of both a 3D model and the living Drosophila embryo (data not shown). "
[Show abstract][Hide abstract] ABSTRACT: Ventral furrow formation in Drosophila is the first large-scale morphogenetic movement during the life of the embryo, and is driven by co-ordinated changes in the shape of individual epithelial cells within the cellular blastoderm. Although many of the genes involved have been identified, the details of the mechanical processes that convert local changes in gene expression into whole-scale changes in embryonic form remain to be fully understood. Biologists have identified two main cell deformation modes responsible for ventral furrow invagination: constriction of the apical ends of the cells (apical wedging) and deformation along their apical-basal axes (radial lengthening/shortening). In this work, we used a computer 2D finite element model of ventral furrow formation to investigate the ability of different combinations of three plausible elementary active cell shape changes to bring about epithelial invagination: ectodermal apical-basal shortening, mesodermal apical-basal lengthening/shortening and mesodermal apical constriction. We undertook a systems analysis of the biomechanical system, which revealed many different combinations of active forces (invagination mechanisms) were able to generate a ventral furrow. Two important general features were revealed. First that combinations of shape changes are the most robust to environmental and mutational perturbation, in particular those combining ectodermal pushing and mesodermal wedging. Second, that ectodermal pushing plays a big part in all of the robust mechanisms (mesodermal forces alone do not close the furrow), and this provides evidence that it may be an important element in the mechanics of invagination in Drosophila.
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