Geometric control of active flows

The development of an organism starting from a fertilized egg involves the self-organized formation of patterns and the generation of shape. Patterns and shapes are characterized by their geometry, i.e. angles and distances between features. In this thesis, we set out to understand how the given geometry of pattern and shape of a living system feeds back into the evolution of this geometry. We focus on two fundamental developmental processes: axis specification and gastrulation. Both processes rely on the directed movements of cells and molecules driven by molecular force generation. Here, we ask how the geometry of an embryo guides such active flows. Active flows are often confined to the surface of a cell or embryo which is usually curved. We use the hydrodynamic theory of active surfaces to investigate how this curvature impacts on flows that are driven by patterns of mechanical activity. Using a minimal model of the cell cortex, we find that active cortical stresses can drive a rotation of the cell that aligns the chemical pattern of the stress regulator with the geometry of the cell surface. In particular, we find that active tension in the cytokinetic ring ensures that a cell divides along its longest axes, a common phenomenon known as Hertwig’s rule. As a consequence, the body axes of the C. elegans embryo are aligned with the geometry of the egg shell. We next set out to understand the impact of surface geometry on flows and patterns in more complex geometries. We focus in particular on localized sources of mechanical activity in curved fluid films. Such active particles act as sensors of the surface geometry, as the viscosity relates the local flow field to the large-scale geometry of the fluid film. We find that the impact of an anisotropic surface geometry on the flow field can generally be understood in terms of effective gradients of friction and viscosity. With this, we show that contractile points in a fluid film are attracted by protrusions and saddl