Parameter Identifiability in PDE Models of Fluorescence Recovery After Photobleaching

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 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.

Appendix A: Derivation of the Sensitivity Equations

We provide details on deriving the sensitivity equations associated with PDE model system (1). Here, we show the derivation of the PDE equations for the sensitivities with respect to the diffusion coefficient D, since the other sensitivities can be derived in a similar way.

We first consider the PDE for the free protein concentration:

$$\begin{aligned} \frac{\partial f}{\partial t}&= D \Delta f -\beta _2 f + \beta _1 c \,. \end{aligned}$$

(29)

We denote $f_D=\frac{\partial f}{\partial D}$ and $c_D=\frac{\partial c}{\partial D}$. We differentiate equation (29) with respect to the diffusion coefficient D and assume that the protein concentrations are smooth and thus have continuous partial derivatives. Applying the chain rule yields:

$$\begin{aligned} \frac{\partial }{\partial D}\left( \frac{\partial f}{\partial t}\right) = \frac{\partial f_D}{\partial t}&= \frac{\partial }{\partial D}(D \Delta f) - \frac{\partial }{\partial D}(\beta _2 f) + \frac{\partial }{\partial D}(\beta _1 c) \nonumber \\&= (f_{xx}+f_{yy}) + D \frac{\partial }{\partial D}\left( f_{xx}+f_{yy}\right) - \beta _2 f_D + \beta _1 c_D \nonumber \\&= \Delta f + D \Delta f_D - \beta _2 f_D + \beta _1 c_D \,. \end{aligned}$$

(30)

Similarly, consider the PDE for the bound protein concentration:

$$\begin{aligned} \frac{\partial c}{\partial t}&= \beta _2 f - \beta _1 c\,. \end{aligned}$$

(31)

Differentiating this equation with respect to D yields the sensitivity equation:

$$\begin{aligned} \frac{\partial }{\partial D}\left( \frac{\partial c}{\partial t}\right) = \frac{\partial c_D}{\partial t}&= \beta _2 f_D - \beta _1 c_D \,. \end{aligned}$$

(32)