Cells display versatile responses to mechanical inputs and recent studies have identified the mitogen-activated protein kinase (MAPK) cascades mediating the biological effects observed upon mechanical stimulation. Although, MAPK pathways can act insulated from each other, several mechanisms facilitate the crosstalk between the components of these cascades. Yet, the combinatorial complexity of potential molecular interactions between these elements have prevented the understanding of their concerted functions. To analyze the plasticity of the MAPK signaling network in response to mechanical stress we performed a non-saturating epistatic screen in resting and stretched conditions employing as readout a JNK responsive dJun-FRET biosensor. By knocking down MAPKs, and JNK pathway regulators, singly or in pairs in Drosophila S2R+ cells, we have uncovered unexpected regulatory links between JNK cascade kinases, Rho GTPases, MAPKs and the JNK phosphatase Puc. These relationships have been integrated in a system network model at equilibrium accounting for all experimentally validated interactions. This model allows predicting the global reaction of the network to its modulation in response to mechanical stress. It also highlights its context-dependent sensitivity.