  Epithelial tissues play a fundamental role in various morphogenetic events
during development and early embryogenesis. Although epithelial monolayers are
often modeled as two-dimensional (2D) elastic surfaces, they distinguish
themselves from conventional thin elastic plates in three important ways - the
presence of an apical-basal polarity, spatial variability of cellular
thickness, and their nonequilibrium active nature. Here, we develop a minimal
continuum model of a planar epithelial tissue as an active elastic material
that incorporates all these features. We start from a full three-dimensional
(3D)description of the tissue and derive an effective 2D model that captures,
through the curvature of the apical surface, both the apical-basal asymmetry
and the spatial geometry of the tissue. Crucially, variations of active
stresses across the apical-basal axis lead to active torques that can drive
curvature transitions. By identifying four distinct sources of activity, we
find that bulk active stresses arising from actomyosin contractility and growth
compete with boundary active tensions due to localized actomyosin cables and
lamellipodial activity to generate the various states spanning the morphospace
of a planar epithelium. Our treatment hence unifies 3D shape deformations
through the coupled mechanics of apical curvature change and in-plane
expansion/contraction of substrate-adhered tissues. Finally, we discuss the
implications of our results for some biologically relevant processes such as
tissue folding at the onset of lumen formation.
