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Cellular mechanics plays an important role in epithelial morphogenesis, aprocess wherein cells reshape and rearrange to produce tissue-scaledeformations. However, the study of tissue-scale mechanics is impaired by thedifficulty of direct measurement of stress in-vivo. Alternative, image-basedinference schemes aim to estimate stress from snapshots of cellular geometrybut are challenged by sensitivity to fluctuations and measurement noise as wellas the dependence on boundary conditions. Here we overcome these difficultiesby introducing a new variational approach - the Geometrical Variation Method(GVM) - which exploits the fundamental duality between stress and cellulargeometry that exists in the state of mechanical equilibrium of discretemechanical networks that approximate cellular tissues. In the GeometricalVariation Method, the two dimensional apical geometry of an epithelial tissueis approximated by a 2D tiling with Circular Arc Polygons (CAP) in which thearcs represent intercellular interfaces defined by the balance of local linetension and pressure differentials between adjacent cells. We take advantage oflocal constraints that mechanical equilibrium imposes on CAP geometry to definea variational procedure that extracts the best fitting equilibriumconfiguration from images of epithelial monolayers. The GVM-based stressinference algorithm has been validated by the comparison of the predictedcellular and mesoscopic scale stress and measured myosin II patterns in theepithelial tissue during Drosophila embryogenesis. GVM prediction of mesoscopicstress tensor correlates at the 80% level with the measured myosin distributionand reveals that most of the myosin II activity is involved in a staticinternal force balance within the epithelial layer. Lastly, this study providesa practical method for non-destructive estimation of stress in live epithelialtissues.

Geometry of epithelial cells provides a robust method for image based inference of stress within tissues