We present a computational framework based on the use of Partial Differential Equations-constrained optimization techniques to model the diffusion of fluorescently tagged molecules on the yeast Endoplasmic Reticulum. We aim at investigating some features of the photobleaching and diffusion of fluorescently tagged molecules on the yeast endoplasmic reticulum (ER) during cell division. We present a computational framework based on Partial Differential Equations-constrained optimization techniques. The optimality conditions are derived and a descent gradient algorithm is used for estimating the diffusion parameters in the cell compartments. With the aid of numerical simulations based on finite element formulations, we support the barrier index conclusions suggesting, based on experimental investigations, the presence of a diffusion barrier in the ER membranes between the two cell domains. A sample result for a realistic geometry of the yeast ER is presented. A relationship between the size ratio between the mother and bud compartments and the barrier index ratio is finally provided.