The γ-chain receptor dimerizes with complexes of the cytokines interleukin-2 (IL-2), IL-4, IL-7, IL-9, IL-15, and IL-21 and their corresponding "private" receptors. These cytokines have existing uses and future potential as immune therapies because of their ability to regulate the abundance and function of specific immune cell populations. Here, we build a binding reaction model for the ligand-receptor interactions of common γ-chain cytokines, which includes receptor trafficking dynamics, enabling quantitative predictions of cell-type-specific response to natural and engineered cytokines. We then show that tensor factorization is a powerful tool to visualize changes in the input-output behavior of the family across time, cell types, ligands, and concentrations. These results present a more accurate model of ligand response validated across a panel of immune cell types as well as a general approach for generating interpretable guidelines for manipulation of cell-type-specific targeting of engineered ligands.