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Geometry can provide long-range mechanical guidance for embryogenesis
Fig 2
Myosin distribution during GB extension.
(a) Fluorescently labeled myosin in the GB and midline over a ventral region early in GB extension [5]. The myosin is significantly denser along DV-oriented cell junctions (y direction) than along AP-oriented ones (x direction). This planar polarisation can be quantified [11, 19, 41]. (b) Sketch of the geometry of the entire embryo with the planar-polarised GB region (green) and the isotropically contracting PMG region (red) [5]. The region shown in panel a is shown from below. Isotropic contraction is assumed to be linked with an isotropic action of myosin, thus σa is an isotropic tensor in the PMG, whereas planar polarisation results in an anisotropic prestress σa [21], whose orientation we take as eDV ⊗ eDV, where eDV is a tangential unit vector orthogonal to the main axis of the embryo. (c) Sketch of the different pools of myosin present at the cell apices. Junctional myosin is associated with cell-cell junctions, and may form supracellular cables. Medial myosin is apical myosin not associated with junctions. (d) Tangential apical stresses in an arbitrary region of the GB. According to the constitutive relation, Eq (2), the (opposite of) viscous stress 
and mechanical stress σ need to balance the myosin prestress σa in both AP and DV directions. Since myosin prestress is zero along AP, the mechanical stress is equal to the viscous stress in this direction, 
, thus AP tension results in extension. In the DV direction, we have 
, resulting in a combination of DV tension and contraction (convergence). The global mechanical balance, Eq (1), has to be solved in order to calculate σ and 
. (e)–(h) Tissue strain, cell intercalation and cell shape change schematics (e) Initial cell arrangement, with planar-polarised myosin along DV-oriented cell-cell junctions (vertical) and the definition of a region of interest to track tissue deformation (hashed area) (f) A combination of cell intercalation (cells marked 1 and 2 are now neighbours) and cell shape and area changes leads to a tissue-scale deformation [15]. The region of interest has now a reversed aspect ratio along AP and DV, and has changed area. (g) A different combination of these cell-scale events (here no intercalation but more extensive pure shear of single cells and same area change) can lead to the same tissue-scale deformation. (h) Tissue scale resistance to deformation can be quantified by two numbers, a shear viscosity η and a second viscosity ηb corresponding to the additional resistance to area variations. Planar-polarised myosin prestress can be retained at tissue scale as an anisotropic prestress tensor σa.
doi: https://doi.org/10.1371/journal.pcbi.1005443.g002