During development, cells consume energy, divide, rearrange, and die. Bulk properties such as viscosity and elasticity emerge from cell-scale mechanics and dynamics. Order appears, for example in patterns of hair outgrowth, or in the predominately hexagonal pattern of cell boundaries in the wing of a fruit fly. In the past fifty years, much progress has been made in understanding tissues as living materials. However, the physical mechanisms underlying tissue-scale behaviour are not completely understood. Here we apply theories from statistical physics and fluid dynamics to understand mechanics and order in two-dimensional tissues. We restrict our attention to the mechanics and dynamics of cell boundaries and vertices, and to planar polarity, a type of long-ranged order visible in anisotropic patterns of proteins and hair outgrowth. Our principle tool for understanding mechanics and dynamics is a vertex model where cell shapes are represented using polygons. We analytically derive the ground-state diagram of this vertex model, finding it to be dominated by the geometric requirement that cells be polygons, and the topological requirement that those polygons tile the plane. We present a simplified algorithm for cell division and growth, and furthermore derive a dynamic equation for the vertex model, which we use to demonstrate the emergence of quasistatic behaviour in the limit of slow growth. All our results relating to the vertex model are consistent with and build off past calculations and experiments. To investigate the emergence of planar polarity, we develop quantification methods for cell flow and planar polarity based on confocal microscope images of developing fly wings. We analyze cell flow using a velocity gradient tensor, which is uniquely decomposed into terms corresponding to local compression, shear, and rotations. We argue that a pattern in an inhomogeneously flowing tissue will necessarily be reorganized, motivating a hydrodynamic theory of polarity reorientation. Using such a coarse-grained theory of polarity reorientation, we show that the quantified patterns of shear and rotation in the wing are consistent with the observed polarity reorganization, and conclude that cell flow reorients planar polarity in the wing of the fruit fly. Finally, we present a cell-scale model of planar polarity based on the vertex model, unifying the themes of this thesis.

Developing tissues are out-of-equilibrium systems that grow and reshape to form organs in adult animals. They are typically composed of a large number of cells. The constitutive cells of a tissue perform different roles in tissue development and contribute to the overall tissue shape changes. In this thesis, we construct a hydrodynamic theory of developing epithelial tissues. We use it to investigate the developing wing of the fruit fly Drosophila melanogaster. This theory relates the coarse-grained cell scale properties to the large-scale tissue flows. We explicitly account for the active cellular processes in the tissue that drive tissue flows. In our description of the tissue, we also include the memory effects that are necessary to account for the experimental observations. These memory effects have a significant influence on the tissue rheology. Using this hydrodynamic theory we analyze shear flow in a developing fruit fly wing tissue. We find that the active cellular processes contribute to overall tissue flows and that memory effects are present in the wing tissue. We investigate consequences of these findings on the rheology of tissue shear flow. We find that the memory effects give rise to an inertial response that leads to oscillations in the tissue but it does not stem from the wing mass. Finally, we describe how the tissue rheology is affected by different boundary conditions. We then investigate the area changes during the pupal wing development and we construct a mechanosensitive model for the cell extrusion rate in the pupal wing. Analysis of cell extrusions in the context of this model also allows us to extract information about the cell division properties. Boundary connections between the wing tissue and surrounding cuticle are crucial for the proper development of the pupal wing. A dumpy mutant wing is strongly misshaped during the pupal wing morphogenesis. We use a simple model for the wing to show that the dumpy mutant wing can be described as a wild type wing with compromised boundary conditions. Finally, we analyze cell properties and tissue flows in a developing wing disc epithelium. Motivated by the observation of radially oriented active T1 transitions in the wing disc epithelium, we use the hydrodynamic theory to investigate the influence of such T1 transitions on stresses in the tissue. We show that sufficiently strong radially oriented active T1 transitions can contribute to the control of the tissue size. Results obtained in this thesis extend our understanding of the fly wing tissue rheology and the role of internal and external forces in the proper shaping of the wing epithelium. The hydrodynamic theory we use to describe the fly wing development provides a set of phenomenological parameters that characterize the tissue mechanics and can be experimentally measured. Therefore, we expect that future research will include and extend the hydrodynamic theory presented in this thesis.