Interleukin-2 (IL2) is a growth factor for several immune cells and its function depends on its binding to IL2Rs in the cell membrane. The most accepted model for the assembling of IL2-IL2R complexes in the cell membrane is the Affinity Conversion Model (ACM). This model postulates that IL2R receptor association is sequential and dependent on ligand binding. Most likely free IL2 binds first to IL2Rα, and then this complex binds to IL2Rβ, and finally to IL2Rγ (γc). However, in previous mathematical models representing this process, the binding of γc has not been taken into account. In this work, the quantitative contribution of the number of IL2Rγ chain to the IL2-IL2R apparent binding affinity and signaling is studied. A mathematical model of the affinity conversion process including the γ chain in the dynamic, has been formulated. The model was calibrated by fitting it to experimental data, specifically, Scatchard plots obtained using human cell lines. This paper demonstrates how the model correctly explains available experimental observations. It was estimated, for the first time, the value of the kinetic coefficients of IL2-IL2R complexes interaction in the cell membrane. Moreover, the number of IL2R components in different cell lines was also estimated. It was obtained a variable distribution in the number of IL2R components depending on the cell type and the activation state. Of most significance, the study predicts that not only the number of IL2Rα and IL2Rβ, but also the number of γc determine the capacity of the cell to capture and retain IL2 in signalling complexes. Moreover, it is also showed that different cells might use different pathways to bind IL2 as consequence of its IL2R components distribution in the membrane.