Homework 8.3: Parameter estimates from FRAP curves (50 pts)¶
Re-read the problem statement for homework 5.1. Recall that in that problem, you extracted fluorescence intensities from a bleached region in a FRAP experiment. You will use these data to obtain maximum likelihood estimates for the physical parameters of interest. As a reminder, the mathematical model, based on diffusion dynamics and chemical exchange, gives the normalized mean intensity of the bleach region as
\begin{align} I_\mathrm{norm}(t) \equiv \frac{I(t)}{I_0} &= f_f\left(1 - f_b\,\frac{4 \mathrm{e}^{-k_\mathrm{off}t}}{d_x d_y}\,\psi_x(t)\,\psi_y(t)\right), \end{align}
where
\begin{align} \psi_i(t) &= \frac{d_i}{2}\,\mathrm{erf}\left(\frac{d_i}{\sqrt{4Dt}}\right) -\sqrt{\frac{D t}{\pi}}\left(1 - \mathrm{e}^{-d_i^2/4Dt}\right), \end{align}
\(I_0\) is the mean fluorescence of the bleached region immediately before photobleaching, \(d_x\) and \(d_y\) are the extent of the photobleached box in the \(x\)- and \(y\)-directions, \(f_b\) is the fraction of fluorophores that were bleached in the bleach region, and \(f_f\) is the fraction of fluorescent species remaining in the entire egg after bleaching. Here, \(\mathrm{erf}(x)\) is the error function. Note that this function is defined such that the photobleaching event occurs at time \(t = 0\).
So, we have four parameters to use in regression, the physical parameters of interest, the diffusion coefficient \(D\) and the chemical rate constant for detachment from the membrane \(k_\mathrm{off}\), and \(f_f\) and \(f_b\).
Assume each of the eight experiments are independent. Obtain MLE estimates, with confidence intervals, for the parameters for each of the eight experiments. You might want to make a plot of all eight values of \(D\) and \(k_\mathrm{off}\).
You should use your own results from HW 5.1 in this problem. However, if (and only if) you were unable to obtain mean fluorescence data from the images, you may usethis data setfrom the HW 5.1 solutions.