A problem of cell-to-cell communication by diffusible ligands is analyzed for the case when cells are distributed in three dimensions and diffusible ligands are secreted by cells and reversibly bind to cell surface receptors. Following its binding to a receptor, the ligand can either dissociate and be released back in the medium or be absorbed by the cell in a process that is called internalization. Using an effective medium approximation, we derive analytical expressions that characterize the time and length scales associated with the ligand trajectories leading to internalization. We discuss the applicability of our approximation and illustrate the application of our results to a specific cellular system
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