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    "# Homework 10.1: Parameter estimates from FRAP curves (Not graded: 0 pts)\n",
    "\n",
    "[Dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/frap_image_processing_results.csv)\n",
    "\n",
    "<hr />"
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    "For this problem, you will perform analysis of data from a fluorescence recovery after photobleaching (FRAP) experiment. The data set comes from [Nate Goehring](https://goehringlab.crick.ac.uk). The data are taken of a *C. elegans* one-cell embryo expressing a GFP fusion to the PH domain of Protein Lipase C delta 1 (PH-PLCd1). This domain binds PIP2, a lipid enriched in the plasma membrane. By using FRAP, we can investigate the dynamics of diffusion of the PH-PLCd1/PIP2 complex on the cell membrane, as well as the binding/unbinding dynamics of PH-PLCd1. The the FRAP experiment, a square patch of the membrane is exposed to intense light, thereby photobleaching the fluorescent molecules. If we quantify how the fluorescence returns to that region, we can infer the diffusion coefficient of the PH-PLCd1/PIP2 complex as well as the binding rate of the two molecules. These are two important parameters for quantifying this system.\n",
    "\n",
    "If $c$ is the concentration of the PH-PLDd1/PIP2 complex on the membrane and $c_\\mathrm{cyto}$ is the concentration of PH-PLCd1 in the cytoplasm (assumed to be spatially uniform since diffusion in the cytoplasm is very fast), the dynamics are described by a reaction-diffusion equation.\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\partial c}{\\partial t} = D\\left(\\frac{\\partial^2 c}{\\partial x^2} + \\frac{\\partial^2 c}{\\partial y^2}\\right) + k_\\mathrm{on} c_\\mathrm{cyto} - k_\\mathrm{off} c.\n",
    "\\end{align}\n",
    "\n",
    "Here, $k_\\mathrm{on}$ and $k_\\mathrm{off}$ are the phenomenological rate constants for binding and unbinding to PIP2 on the membrane, and $D$ is the diffusion coefficient for the PH-PLCd1/PIP2 complex on the membrane.\n",
    "\n",
    "In [their paper](http://dx.doi.org/10.1016/j.bpj.2010.08.033), the authors discuss techniques for analyzing the data taking into account the fluorescence recovery of the bleached region in time and space. If $I(t)$ is the mean fluorescence of the bleached region and $I_0$ is the mean fluorescence of the bleached region immediately before photobleaching, we have, as derived in the paper,\n",
    "\n",
    "\\begin{align}\n",
    "I(t) &= \\left\\{\\begin{array}{lll}\n",
    "I_0 & & t < 0 ,\\\\[1em]\n",
    "I_0 \\left( \n",
    "1 - f_b\\,\\frac{4 \\mathrm{e}^{-k_\\mathrm{off}t}}{d_x d_y}\\,\\psi_x(t)\\,\\psi_y(t)\\right) & & t \\ge 0,\n",
    "\\end{array}\n",
    "\\right.\\\\[1em]\n",
    "\\text{where } \\psi_i(t) &= \\frac{d_i}{2}\\,\\mathrm{erf}\\left(\\frac{d_i}{\\sqrt{4Dt}}\\right)\n",
    "-\\sqrt{\\frac{D t}{\\pi}}\\left(1 - \\mathrm{e}^{-d_i^2/4Dt}\\right),\n",
    "\\end{align}\n",
    "\n",
    "where $d_x$ and $d_y$ are the extent of the photobleached box in the $x$- and $y$-directions, $f_b$ is the fraction of fluorophores in the bleach region that were bleached, and $\\mathrm{erf}(x)$ is the [error function](http://en.wikipedia.org/wiki/Error_function). For this experiment, $d_x = d_y = 5.52$ µm. Note that this function is defined such that the photobleaching event occurs at time $t = 0$.\n",
    "\n",
    "We measure $I(t)$, $d_x$, and $d_y$. We could also measure $f_b$ as\n",
    "\n",
    "\\begin{align}\n",
    "f_b \\approx 1 - \\frac{I(0^+)}{I_0},\n",
    "\\end{align}\n",
    "\n",
    "though we will consider this a parameter to estimate in our analysis. In practice, the normalized fluorescent recovery does not go all the way to unity. This is because the FRAP area is a significant portion of the membrane, and we have depleted fluorescent molecules. We should thus adjust our equation to be\n",
    "\n",
    "\\begin{align}\n",
    "I(t) &= \\left\\{\\begin{array}{lll}\n",
    "I_0 & & t < 0 ,\\\\[1em]\n",
    "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) & & t \\ge 0,\n",
    "\\end{array}\n",
    "\\right.\\\\[1em]\n",
    "\\text{where } \\psi_i(t) &= \\frac{d_i}{2}\\,\\mathrm{erf}\\left(\\frac{d_i}{\\sqrt{4Dt}}\\right)\n",
    "-\\sqrt{\\frac{D t}{\\pi}}\\left(1 - \\mathrm{e}^{-d_i^2/4Dt}\\right),\n",
    "\\end{align}\n",
    "\n",
    "where $f_f$ is the fraction of fluorescent species in the entire embryo that remain unbleached. So, we have five parameters to use in regression, the physical parameters of interest, $D$ and $k_\\mathrm{off}$, and $I_0$, $f_f$, and $f_b$.\n",
    "\n",
    "I have processed images from the data set. The results may be downloaded here: [https://s3.amazonaws.com/bebi103.caltech.edu/data/frap_image_processing_results.csv](https://s3.amazonaws.com/bebi103.caltech.edu/data/frap_image_processing_results.csv). There are eight FRAP experiments in total. 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 $k_\\mathrm{off}^*$ with their confidence regions."
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