To understand how long-range patterning gradients are interpreted at the cellular level, we investigate how a gradient of expression of the Four-jointed kinase specifies planar polarised distributions of the cadherins Fat and Dachsous in the Drosophila wing. We use computational modelling to test different scenarios for how Four-jointed might act and test the model predictions by employing fluorescence recovery after photobleaching as an in vivo assay to measure the influence of Four-jointed on Fat-Dachsous binding. We demonstrate that in vivo, Four-jointed acts both on Fat to promote its binding to Dachsous and on Dachsous to inhibit its binding to Fat, with a bias towards a stronger effect on Fat. Overall, we show that opposing gradients of Fat and Dachsous phosphorylation are sufficient to explain the observed pattern of Fat–Dachsous binding and planar polarisation across the wing, and thus demonstrate the mechanism by which a long-range gradient is interpreted. DOI: http://dx.doi.org/10.7554/eLife.05789.001, eLife digest Epithelial cells form sheets that line the body surfaces and internal cavities of animals—such as the skin and the lining of the gut. Certain structures on the surface of epithelial cell sheets—for example scales, hair, and feathers—are often all orientated in a particular direction. Epithelial cells with structures organised like this are described as being ‘planar polarised’. Different proteins work together to set up planar polarity in a sheet of epithelial cells. Dachsous and Fat are two proteins that are found in the cell membranes of epithelial cells, including in the wings of the fruit fly Drosophila. These proteins bind to each other and link a cell to its neighbour. Dachsous and Fat accumulate on opposing sides of each cell: Fat accumulates on the side closest to the fly's body, and Dachsous builds up on the side closest to the wing tip. This pattern provides directional cues that help orientate surface structures, and the pattern is established, in part, by the activity of an enzyme called Four-jointed. Four-jointed adds phosphate groups onto Dachsous and Fat. The activity of the Four-jointed enzyme forms a gradient along a developing wing: levels are low near the fly's body, and high at the wing tip. Previous experiments performed on cells grown in the laboratory showed that when Four-jointed adds phosphate groups to Fat and Dachsous, it prevents Dachsous from binding to Fat. However, it also makes Fat more able to bind to Dachsous. These opposing effects are thought to cause the proteins to accumulate on opposing sides of each cell. However, this has yet to be demonstrated in real tissue, not least because of the technical difficulty of measuring whether Fat-Dachsous binding has occurred in living organisms. Here, Hale et al. overcome this challenge using a method called ‘fluorescence recovery after photobleaching’ (or FRAP) to measure Fat and Dachsous binding in the epithelial cells in the developing Drosophila wing. Combining these experimental results with a computational model confirmed the findings of previous laboratory studies: that Four-jointed makes it easier for Fat to bind to Dachsous, and makes it more difficult for Dachsous to bind to Fat. The opposing effects on the activity of Fat and Dachsous that result from the Four-jointed gradient in the developing wing are able to fully explain the observed patterns of Fat-Dachsous binding and of planar polarisation across the wing. Overall, Hale et al. demonstrate how a gradient of protein activity that spans many cells is sensed and interpreted by individual cells to establish planar polarity. However, exactly how the phosphate groups added to Dachsous and Fat by Four-jointed modifies how they bind to each other remains a question for future work. DOI: http://dx.doi.org/10.7554/eLife.05789.002