  Identifying unique parameters for mathematical models describing biological
data can be challenging and often impossible. Parameter identifiability for
partial differential equations models in cell biology is especially difficult
given that many established \textit{in vivo} measurements of protein dynamics
average out the spatial dimensions. Here, we are motivated by recent
experiments on the binding dynamics of the RNA-binding protein PTBP3 in RNP
granules of frog oocytes based on fluorescence recovery after photobleaching
(FRAP) measurements. FRAP is a widely-used experimental technique for probing
protein dynamics in living cells, and is often modeled using simple
reaction-diffusion models of the protein dynamics. We show that current methods
of structural and practical parameter identifiability provide limited insights
into identifiability of kinetic parameters for these PDE models and
spatially-averaged FRAP data. We thus propose a pipeline for assessing
parameter identifiability and for learning parameter combinations based on
re-parametrization and profile likelihoods analysis. We show that this method
is able to recover parameter combinations for synthetic FRAP datasets and
investigate its application to real experimental data.
