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    "# Problem 3.4: Modeling inducers in a toggle switch\n",
    "\n",
    "<hr>"
   ]
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    "*This problem is still a draft.*\n",
    "\n",
    "In our analysis of the Gardner and Collins synthetic toggle switch, we did not include a way to represent the addition of the inducer signals. Such signals typically function by decreasing a receptor's affinity for its target binding site, in other words by increasing the value of the Hill repression constant $k$. In the nondimensionalized system, we see that $k_x$ (and correspondingly, $k_y$) appear both in the dimensionless concentration $\\tilde{x}$ and in the dimensionless production term $\\tilde{\\beta}_x$.\n",
    "    \n",
    "Create an interactive plot representing the following scenario:\n",
    "\n",
    "* The circuit is built with parameters $\\beta, \\gamma, n$ fixed to be $\\beta = 10$, $\\gamma = 1$, $n=10$.\n",
    "* The plot has sliders for $k_x$ and $k_y$, which represent the presence of inducer signal to inactivate the repressors $X$ and $Y$, respectively. (Note that this representation assumes that the inducer's action is linearly proportional to its concentration, which is typically not true but gives the right qualitative result).\n",
    "* The plot shows how the nullclines, in dimensionless concentration space, change as you add and remove the inducer signals.\n",
    "\n",
    "What insights did you gain from the visualization?"
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   "source": [
    "<br />"
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