Schematic of the multi-scale model of the 2Dsection of the <i>Drosophila</i> embryo.

<p>(<b>a</b>) Biological architecture of the <i>Drosophila</i> single epithelial cell. (<b>b</b>) Each architectural constituent of the epithelial cell plays a structural role during cell shape changes. Such roles can be either active or passive, depending on whether that constituent is able to generate force or it is deformed by forces actively generated by neighbouring elements. Actin-myosin forces are known to be active systems for force generation whereas microtubules are known to maintain stiffness. Apical junction complexes, instead, keep cell junctions connected and, therefore, transmit forces from one cell to other cells in the epithelium. The inner cytoplasm is a viscous incompressible fluid that resists and responds to cell compression by exerting an inner pressure. (<b>c</b>) In order to build a finite element model of the single epithelial cell, the phenotypical effects of the complex force fields originating from the molecular level can be thought as an equivalent field of net forces acting along the cell edges. The incompressibility of the inner cytoplasm can be simulated by enforcing the surface of the region enclosed by the cell membrane to remain constant in time. (<b>d</b>) Individual cells (green) are broken into quadrilateral finite elements (blue), which are used to calculate the forces generated by their passive (viscous) components during each time step t<i>_i</i> of the simulation. Elements are joined to each other at nodes, and the set of forces (blue arrows) that would be needed to drive a particular set of nodal motions <b>u</b>_i (magenta arrows) is denoted by <b>f</b>_i. These passive forces <b>f</b>_i and their corresponding displacements <b>u</b>_i are related by Eqn. (1). The forces generated by active cellular components are resolved into edge forces which act on both of the nodes that mark their ends. Some of these edge forces are shown (red arrows), as is their vector sum at a representative node (blue arrows). Similar vector sums for each node give rise to the collection of nodal driving forces <b>f</b>_i* which are set to be equal to the passive forces <b>f</b>_i generated by the cytoplasm. Setting these forces equal to each other means that the vectorial driving forces acting at each node are just balanced by viscous resistance from the cytoplasm at that node. (<b>e</b>) A finite element model of the 2D section of the Drosophila embryo is built by matching to <i>in vivo</i> images at the cellular level. 2D geometry has been sized on the <i>in vivo</i> dimensions of the wild type embryo at the reference initial instant t = −10.43 min (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034473#pone-0034473-g001" target="_blank">fig. 1b</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034473#pone-0034473-g002" target="_blank">2a</a>). Mesodermal region covers approximately 60 degrees across the ventral point V and is highlighted in dark green, whereas ectodermal region is highlighted in bright green. The presence of the vitelline membrane has been simulated by constraining the motion of apical nodes unilaterally, in such a way that apical nodes could not displace at distances from the centre of the circular epithelium greater than the radius of the circular vitelline membrane (red circle). The effect of the presence of the inner viscous yolk in the real embryo has been modelled, instead, by imposing a user defined pressure to the basal nodes. The value of the pressure is than chosen in such a way that surface variations of the inner yolk region (orange area) are approximately of the same magnitude of <i>in vivo</i> yolk area variations (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034473#pone.0034473.s001" target="_blank">Fig. S1</a>).