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        "# Determining dissociation constants\n",
        "\n",
        "[Data set 1 download](https://s3.amazonaws.com/bebi103.caltech.edu/data/rasson_dissociation_anisotropy_tidy.csv)\n",
        "\n",
        "<hr>"
      ]
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        "In this problem, we will work with data from a [paper by Rasson and coworkers](https://doi.org/10.1016/j.jmb.2014.09.002). The authors were investigating the biochemistry of Spire-actin interactions. Spire is an actin binding protein that can nucleate actin filaments. In particular, it has four domains (called $S_A$, $S_B$, $S_C$, and $S_D$), which bind monomeric actin. These four domains, acting in concert, can line up actin monomers to help in nucleation. In this problem, we will determine the dissociation constant, $K_d$, describing binding of $S_D$ to monomeric actin.\n",
        "\n",
        "The strategy to determine $K_d$ is to perform a titration experiment and then use parameter estimation with a variate-covariate model to determine $K_d$. Consider the chemical reaction describing $S_D$-actin binding.\n",
        "\n",
        "\\begin{align}\n",
        "\\text{actin}\\cdot S_D \\rightleftharpoons \\text{actin} + S_D,\n",
        "\\end{align}\n",
        "\n",
        "which has dissociation constant $K_d$. (In this system actin does not polymerize and stays monomeric because it is treated with Latrunculin.) Let $c_a$ be the equilibrium concentration of actin and $c_d$ be the equilibrium concentration of $S_D$, and $c_{ad}$ be the equilibrium concentration of bound actin-$S_D$. Then, at equilibrium,\n",
        "\n",
        "\\begin{align}\n",
        "K_d = \\frac{c_a \\, c_d}{c_{ad}}.\n",
        "\\end{align}\n",
        "\n",
        "If we started with a total actin concentration of $c_a^0$ and a total $S_D$ concentration of $c_d^0$, we also have\n",
        "\n",
        "\\begin{align}\n",
        "c_a^0 = c_a + c_{ad}, \\\\[1mm]\n",
        "c_d^0 = c_d + c_{ad},\n",
        "\\end{align}\n",
        "\n",
        "by conservation of mass. With these relations, we can now write $c_{ad}$ in terms of $c_a^0$ and $c_d^0$, which are known quantities (this is what we pipetted into our solution).\n",
        "\n",
        "\\begin{align}\n",
        "K_d &= \\frac{(c_a^0 - c_{ad})(c_d^0 - c_{ad})}{c_{ad}},\\\\[1mm]\n",
        "\\Rightarrow\\;&\\;\\;c_{ad}^2 - (K_d + c_a^0 + c_d^0)c_{ad} + c_a^0 c_d^0 = 0.\n",
        "\\end{align}\n",
        "\n",
        "The solution to this quadratic equation gives $c_{ad}$ as a function of $K_d$. Note that we must choose one of the two roots, the one that is physical. The physical root satisfies $0 < c_{ad} < \\min(c_a^0, c_d^0)$. In this case, it is\n",
        "\n",
        "\\begin{align}\n",
        "c_{ad} = \\frac{1}{2}\\left(K_d + c_a^0 + c_d^0 - \\sqrt{\\left(K_d + c_a^0 + c_d^0\\right)^2 - 4c_a^0c_d^0}\\right).\n",
        "\\end{align}\n",
        "\n",
        "In a titration experiment, we fix $c_d^0$ and vary $c_a^0$, and measure $c_{ad}$ to get a curve. From the curve, we can perform a regression to get $K_d$.\n",
        "\n",
        "The problem with this approach is that we do not have a direct way of measuring $c_{ad}$.  The authors instead employed [fluorescence anisotropy](https://en.wikipedia.org/wiki/Fluorescence_anisotropy). I will not go into the details here of how it works, but will simply say that larger complexes rotate more slowly, and therefore give a higher fluorescence anisotropy signal (which is dimensionless) than do smaller complexes.\n",
        "\n",
        "So, the authors fluorescently tagged $S_D$. We will call this molecule $S_{D^*}$, with concentration $c_{d^*}$. When free in solution, this molecule gives an anisotropy signal of $r_f$. When bound to actin, it gives an anisotropy signal of $r_b$. So, the total anisotropy signal we detect is\n",
        "\n",
        "\\begin{align}\n",
        "r = \\frac{1}{c_{d^*}^0}\\,\\left(r_f c_{d^*} + r_b c_{ad^*}\\right).\n",
        "\\end{align}\n",
        "\n",
        "Clearly, when all $S_{D^*}$ is free, the anisotropy signal is $r_f$ and when all $S_{D^*}$ is bound to actin, the signal is $r_b$. Remembering conservation of mass, $c_{d^*} = c_{d^*}^0 - c_{ad^*}$, we have\n",
        "\n",
        "\\begin{align}\n",
        "r = \\frac{1}{c_{d^*}^0}\\,\\left(r_f (c_{d^*}^0 - c_{ad^*}) + r_b c_{ad^*}\\right)\n",
        "= r_f  + \\frac{r_b-r_f}{c_{d^*}^0}\\, c_{ad^*}.\n",
        "\\end{align}\n",
        "\n",
        "Now, returning to our equilibrium expression, we have\n",
        "\n",
        "\\begin{align}\n",
        "c_{ad^*} = \\frac{1}{2}\\left(K_d^* + c_a^0 + c_{d^*}^0 - \\sqrt{\\left(K_d^* + c_a^0 + c_{d^*}^0\\right)^2 - 4c_a^0c_{d^*}^0}\\right),\n",
        "\\end{align}\n",
        "\n",
        "so we can write the measured anisotropy $r$ as a function of $K_d^*$ and the known quantities $c_a^0$ and $c_{d^*}^0$. Note that we now have three parameters in our mathematical model, $K_d^*$, $r_f$, and $r_b$, since the latter two are not known *a priori*.\n",
        "\n",
        "Your job is to get a parameter estimate for $K_d^*$ with a confidence interval."
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