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(Left) Schematic representation a two-dimensional tissue with a quasi one-dimensional distribution of ligand production. Diffusible ligand is produced from the spatially localized source, diffuses through the tissue, binds to cell surface receptors, and can be internalized by cells. (Right) An expertimental example that fits this model. A spatially uniform arrangement of cells in the early Drosophila embryo (top panel, gray), spatially restricted source of ligand production (middle panel, red), and the resulting activation of the intracellular signaling pathway (bottom panel, green). The ligand binds to and activates receptors which are uniformly expressed throughout the embryo.
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Single-exponential approximation of the concentration dynamics. The four panels present concentration profiles of mobile and immobile species at different times: , 2, 4, and 10, where is given in Eq. (4.9) . The profiles are normalized to the corresponding steady-state values at x = 0. The red and blue lines represent the mobile and immobile species, respectively. The black lines are the normalized steady-state profile profiles, exp (−x / λ eff ), where λ eff is given in Eq. (4.20) . Solid and dashed curves are exact solutions for the concentration profiles obtained numerically and their single-exponential counterparts, Eq. (5.1) , respectively. The quality of the single-exponential approximation improves with time. The plots show the dependences for the following choice of the problem parameters: D = Q = 1, , and .
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