Parameter identifiability in PDE models of fluorescence recovery after photobleaching

TL;DR

A pipeline for assessing parameter identifiability and for learning parameter combinations based on re-parametrization and profile likelihoods analysis is proposed and it is shown that this method is able to recover parameter combinations for synthetic FRAP datasets and investigate its application to real experimental data.

Abstract

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.

Figures (10)

• Figure 1: A) Schematic of a stage II Xenopus oocyte with RNA granules localizing at the vegetal cortex (bottom) shown in magenta. The black square region is shown magnified on the right, with a cartoon of a FRAP bleach spot. B) The timeline of Fluorescence Recovery After Photobleaching (FRAP) shows bleaching of a small square region (at the bleach time) in a previously-fluoresced region of the cell (at the pre-bleach time). The first postbleach time already shows that non-bleached and fluorescent molecules mix between the fluoresced and bleached regions. C) An image of the vegetal cytoplasm of a Xenopus laevis oocyte expressing fluorescently-labeled PTBP3 (red) in L-bodies is shown, with a $10~\mu$m photobleach square ROI. This image corresponds to the postbleach time point in the cartoon in B. Yellow dashed lines show sample extraction of the fluorescence profiles $C_b(x)$ and $C_b(y)$ in the $x$ and $y$ directions from the postbleach intensity data, as shown in panel B as well. D) Fitted fluorescence postbleach profiles along the $x$ and $y$ directions. The estimated parameters are $\alpha_x=2.33$, $r_x=5.64$ (with $R^2=0.39$) and $\alpha_y=2.72$, $r_y=3.27$ (with $R^2=0.37$).
• Figure 2: Interpretation of profile likelihoods in terms of structural and practical identifiability raue2013joining. A flat likelihood (left) corresponds to structural non-identifiability, a profile that does not decrease to $0$ on one or both sides of the maximum (center) indicates practically non-identifiability, and a profile with a fast decrease to $0$ on both sides of the maximum (right) shows both structural and practical identifiability.
• Figure 3: MCMC DRAM-estimated A) univariate and B) bivariate marginal distributions for noiseless FRAP data generated using Parameter Set 2 in Table \ref{['tab:par_regimes']}. Scale bars in Panel B correspond to the number of sampled points in the MCMC simulation for each parameter pair.
• Figure 4: Profile likelihoods for each interest parameter on the $x$ axis given noiseless FRAP data synthetically generated using model \ref{['eq:diff_pause']} and Parameter Set 2 in Table \ref{['tab:par_regimes']}.
• Figure 5: Subset profiles for each interest rate parameter on the $x$ axis and the corresponding optimized nuisance rate parameter on the $y$ axis given noiseless FRAP data synthetically generated using model \ref{['eq:diff_pause']} and Parameter Set 2 in Table \ref{['tab:par_regimes']}. The true reaction rate parameters are indicated with red circles.