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    "# Problem 5.1: Robustness in a C1-FFL\n",
    "\n",
    "<hr>"
   ]
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    "*This problem is still a draft.*\n",
    "\n",
    "Consider the design principle we discussed in a [previous chapter](04_ffls.ipynb): **The C1-FFL with AND logic displays an on-delay.** We saw that this property of the C1-FFL holds even when the Hill coefficient for the regulation is unity. We might also ask if the delay is robust to variations in the Hill activation constants.\n",
    "\n",
    "As a reminder, the dimensionless dynamical equations for the concentrations of Y and Z from a stimulus X are\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}y}{\\mathrm{d}t} &= \\beta\\,\\frac{(\\kappa x)^{n_{xy}}}{1 + (\\kappa x)^{n_{xy}}} - y, \\\\[1em]\n",
    "\\gamma^{-1}\\frac{\\mathrm{d}z}{\\mathrm{d}t} &= \\frac{x^{n_{xz}} y^{n_{yz}}}{(1 + x^{n_{xz}})\\,(1+ y^{n_{yz}})} - z.\n",
    "\\end{align}\n",
    "\n",
    "To investigate the effect of the Hill activation constants, we need only to vary the dimensionless parameter $\\kappa$, which is the ratio of the Hill activation constant for activation of Z by X to the Hill activation constant for activation of Z by Y; $\\kappa = k_{xz}/k_{yz}$.\n",
    "\n",
    "**a)** Investigate the dynamics of this circuit in response to a step in X concentration to make a robustness statement about on-delay for varying $\\kappa$.\n",
    "\n",
    "**b)** Make a robustness statement about the steady state levels of Z for varying $\\kappa$."
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