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   "source": [
    "## Incoherent Type 4 FFL with AND gate\n",
    "\n",
    "Implement the incoherent type 4 FFL with an AND gate, as described by Mangan & Alon (PNAS, 2003).\n",
    "\n",
    "The parameters are\n",
    "- B_y: basal rate of Y's transcription\n",
    "- Beta_y: maximum rate of Y's transcription\n",
    "- B_z: basal rate of Z's transcription\n",
    "- Beta_z: maximum rate of Z's transcription\n",
    "- K_xy: the activation threshold for X's activation of Y\n",
    "- K_xz: the activation threshold for X's activation of Z\n",
    "- K_yz: the activation threshold for Y's activation of Z\n",
    "- H: the Hill coefficient (exponent) for all regulation\n",
    "- alpha_y: the degradation rate constant for Y\n",
    "- alpha_z: the degradation rate constant for Z\n",
    "- SX: amount of active X in system (X*=SX)\n",
    "- SY: whether or not Y is activated (Y*=Y if SY=1, Y*=0 if SY=0)\n",
    "\n",
    "The model is\n",
    "\\begin{eqnarray*}\n",
    "\\frac{dY}{dt} &=& B_y + \\beta_Y \\frac{1}{1+(X^*/K_{xy})^H} - \\alpha_y Y\\\\\n",
    "\\frac{dZ}{dt} &=& B_z + \\beta_Z \\left(\\frac{(X^*/K_{xz})^H}{1+(X^*/K_{xz})^H}\\right) \\left(\\frac{(Y^*/K_{yz})^H}{1+(Y^*/K_{yz})^H}\\right) - \\alpha_z Z\\\\\n",
    "\\end{eqnarray*}\n",
    "\n",
    "where $X^*=$SX and $Y^*=Y$ if SY=1 and $Y^*=0$ if SY=0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def incoherent_type_4_and_model( t, statevars, params ):\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulate the model\n",
    "\n",
    "Simulate the model. In the paper, they specify that for most simulations,\n",
    "\n",
    "- B_y = 0\n",
    "- Beta_y = 1\n",
    "- B_z = 0\n",
    "- Beta_z = 1\n",
    "- H = 2\n",
    "- alpha_y = 1\n",
    "- alpha_z = 1\n",
    "\n",
    "The activation thresholds are given in the figure captions, as are SX and SY.\n",
    "\n",
    "Simulate with SX=0 for t=0 to t=0.5, then SX=1 for t=0.5 to t=2.5, then SX=0 from t=2.5 to t=6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot the simulation\n",
    "Plot X over time in an upper subplot and Z over time in a lower subplot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "#Write code here"
   ]
  }
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