We present a framework to model and provide numerical evidence for
compartmentalization in the yeast endoplasmic reticulum. Measurement
data is collected and an optimal control problem is formulated as a
regularized inverse problem. To our knowledge, this is the first attempt
in the literature to introduce a PDE-constrained optimization
formulation to study the kinetics of fluorescently labeled molecules in
budding yeast. Optimality conditions are derived and a gradient descent
algorithm allows accurate estimation of unknown key parameters in
different cellular compartments. For the first time, the numerical
results support the barrier index theory suggesting the presence of a
physical diffusion barrier that compartmentalizes the endoplasmic
reticulum by limiting protein exchange between the mother and its
growing bud. We report several numerical experiments on real data and
geometry, with the aim of illustrating the accuracy and efficiency of
the method. Furthermore, a relationship between the size ratio of mother
and bud compartments and the barrier index ratio is provided.